Amitsur's theorem, semicentral idempotents, and additively idempotent semirings

: The article explores research ﬁ ndings akin to Amitsur ’ s theorem, asserting that any derivation within a matrix ring can be expressed as the sum of an inner derivation and a hereditary derivation. In most results related to rings and semirings, Birkenmeier ’ s semicentral idempotents play a crucial role. This article is intended for PhD students, postdocs, and researchers.


Introduction
This article examines results for the derivations of matrix rings related to the well-known theorem of Amitsur for representing a derivation as a sum of an inner and a hereditary derivation.We also consider semicentral idempotents created by Birkenmeier as they are a useful tool for studying derivations.Finally, we review a number of papers on the derivations of some classes of additively idempotent semirings in which Amisur's theorem and semicentral idempotents play a crucial role.This article does not exhaustively explore all possible derivations of rings and semirings.Notably absent are discussions on significant topics such as Jordan derivations, derivations of Lie algebras, and local derivations.In addition, we have opted to exclude papers that merely "generalize" results from ring theory to semirings, as they fall outside the scope of our interest.
Furthermore, we have not delved into the extensive body of work by Rowen and the mathematical community surrounding him, particularly on the generalization of ideas from classical algebraic geometry to the emerging field of additively idempotent semirings.While we acknowledge the importance of numerous papers in this domain, we have selectively marked some for reference and encourage someone from this community to conduct a comprehensive review.
Our research covers aspects of ring and semiring theory, aiming to elucidate the delineations that distinguish it.We will specifically highlight the articles and books employed as references to define the scope of our study: • the excellent survey article on decomposition of matrices as a product of idempotents by Jain and Leroy [1]; • the review article on the derivations of rings by Ashraf et al. [2]; • the scientific monograph of Birkenmeier et al. [3]; • the survey paper of Rachev on endomorphism semirings [4]; • the book of Głazek [5] which is an extensive list of sources in semiring theory; • the survey paper of our colleague Dimitrov [6] concerning derivations of semirings; • the very detailed overview of semiring applications, see Golan [7].
Let M be a ring of matrices over R. It is a well-known fact that the map , where δ is a derivation of R, is a derivation called a hereditary derivation generated by δ.In 1982, Amitsur [20, Theorem 2] proved that an arbitrary derivation of the ring of square × n n matrices M R n ( ) over an associative ring R with identity is a sum of an inner derivation and a hereditary derivation.
In 1983, Nowicki [21] showed a similar result for the so-called special subrings of rings of square × n n matrices over ring.
A particular case of Amitsur's result, when R is an algebra over a field Φ, with ≠ char Φ 2, 3 ( ) , appears in Benkart and Osborn [22].
In 1993, Coelho and Milies [23] proved the following result similar to those in by Amitsur [20] for the ring of upper triangular matrices T R n ( ): This result appeared in 1982 in the last example of Mathis [24].
In 1990, Kezlan [25] obtained the analogous result for an automorphism, which states the following: Theorem 2.2.If R is any commutative ring with unity, then every R-algebra automorphism of T R n ( ) is inner.
In 1995, Jondrup [26] generalized this theorem to rings in which all idempotents are central, and by using the method of generalization, he re-proved the results of Mathis and of Coelho and Milies.In 1951, Dubisch and Perlis [27] studied a new class of matrix rings NT F n ( ) consisting of matrices over a field F whose entries are zeroes on and over the main diagonal.They proved that any automorphism on NT F n ( ) is a product of certain diagonal automorphism, inner automorphism, and nil automorphism.
Kuzucuoğlu and Levchuk [28,29] have investigated the ideals and automorphisms of the ring where K is an associative ring and J is an ideal of K .In 2006, Chun and Park [30] defined the following derivations of the ring NT K n ( ): (1) for each diagonal matrix ] is an inner derivation determined by d of the ring of lower triangular matrices over K and it is called a diagonal derivation, and (2) a derivation . The main result of the study by Chun and Park [30] is Theorem 2.3.An arbitrary derivation of NT K n ( ) is a sum of a diagonal derivation, a hereditary derivation, and a strongly nilpotent derivation.
In 2010, Levchuk and Radchenko [31] generalized the theorem of Chun and Park replacing the strongly nilpotent derivation with a central derivation.(A derivation (or automorphism) of a ring is called central if it acts like the zero (resp.identity) map modulo the center.) Derivations of a matrix ring containing a subring of triangular matrices was described in 2011 by Kolesnikov and Mal'tsev [32] using the results of the study by Levchuk and Radchenko [31].
Derivation of matrix rings consisting of sums of a niltriangular matrix and a matrix over an ideal were studies by Kuzucuoğlu and Sayin [33] in 2017.
As infinite matrices, in 2015, Słowik [34] considered the ring M R Cf ( ) consisting of all infinite matrices over an associative ring R with a finite number of nonzero entries in each column.He also denoted by d A the inner derivation determined by the matrix A and by δ ͠ the hereditary derivation of M R Cf ( ) generated by the derivation δ of R. The first main result is the following: Theorem 2.4.Let R be an associative unital ring.If d is a derivation of M R Cf ( ), then there exists a matrix The second main result is a similar equality for the ring of infinite matrices with finite number of nonzero entries in every row.
In 2017, Hołubowski et al. [35] showed that any derivation of the Lie algebra of infinite strictly upper triangular matrices over a commutative ring is the sum of an inner derivation and a diagonal derivation.
In 2022, Brešar [36] obtained the following result, closely related to Amitsur's idea: Theorem 2.5.Let A be a finite-dimensional algebra over a field F with [ ] for every ∈ x A, then D is the sum of an inner derivation of A and a linear map from A to rad A ( ).
Following the theorem, the author drew two conclusions.
Corollary 2.6.Let A be a finite-dimensional semisimple algebra over a field F with . The following conditions are equivalent for a linear map A local derivation of an algebra A is defined as a linear map x ( ) ( ), see for details Kadison [37].Local automorphisms are defined simi- larly.Note that local automorphisms play an important role in functional analysis (see the study by Larson and Sourour [38]).A standard question is whether local derivations (resp., local automorphisms) are derivations (resp., automorphisms).
Corollary 2.7.Let A be a finite-dimensional semisimple algebra over a field F with ≠ F char 2 ( ) . Then every local inner derivation → D A A : is an inner derivation.

Similar research
Exploring whether there are objects in ring theory that share a research history with well-known objects of the same type is an intriguing aspect worth investigating.Also, in the matrix ring, there is research to present a matrix as a product, a sum, a difference, or linear combinations of idempotent matrices.We leave it to the reader to draw their own conclusions from these investigations.
Over the last 60 years, substantial efforts have been devoted to the examination of idempotent compositions in matrix rings.
In 1966, Howie [39] proved that every transformation of a finite set which is not permutation can be written as a product of idempotents.One year later, Erdos [40] proved that every singular matrix over a field is a product of idempotent matrices.This result was extended to matrices over division rings and Euclidean rings.In many papers, the connection between product decomposition of singular matrices into idempotents and product decomposition of invertible matrices into elementary matrices is considered.An × n n matrix over ring R is called elementary if it is of the form + I ce n i j , where ∈ c R and ≠ i j.In 2014, Salce and Zanardo [41] studied relations between these two decompositions in the setting of commutative integral domains.
In 2018, Cossu et al. [42] proved that the property every invertible × n n matrix over integral domain is a product of elementary matrices holds for important classes of non-Euclidean principal integral domains as coordinate rings of elliptic curves having only one rational point.
In 2019, Cossu and Zanardo [43] proved that an integral domain R such that any singular × 2 2 matrix over R is a product of idempotent matrices must be a Prüfer domain in which every invertible × 2 2 matrix is a product of elementary matrices.
In 2022, Cossu and Zanardo [44] proved that any × 2 2 matrix over the ring of integers of the real quadratic where > d 0 is a square-free integer, with either null row or a null column is a product of idempotents.
We now come to consider idempotent factorizations of matrices over noncommutative rings.
In 2014, Alahmadi et al. [45] provided constructions of idempotents to represent typical singular matrices over a ring (not necessarily commutative) as products of idempotents.The important results are as follows: If A is × 2 2 matrix over local ring with ≠ r ann A .0 ( ) , then A is a product of idempotent matrices.If every × 2 2 invertible matrix over Bézout domain is a product of elementary matrices and diagonal matrices with invertible diagonal entries, then every × n n singular matrix is a product of idempotent matrices.
In 2014, Alahmadi et al. [46] considered various conditions for ring R connected with the decomposition of singular matrices over R as a product of idempotent matrices.
Very recently, in 2024, Vladeva presented a new formula of all semicentral idempotents of upper triangular matrix rings [51].

The studies of Birkenmeier and the mathematical community around him
Most research papers by the group around Birkenmeier explore when the properties of a ring (or a module) are transferred to its various ring extensions (or module extensions).Semicentral idempotents are a key tool for these studies.Therefore, we will review some of the scientific papers where semicentral idempotents play an important role.
In 1983, Birkenmeier [52] defined new idempotent elements of an associative ring R as follows: an idempotent ∈ e R is called a left (resp., right) semicentral idempotent if = exe xe (resp., = exe ex) for all ∈ x R. Birkenmeier et al. [53] developed the theory of generalized triangular matrix representations.Let R be an associative K -algebra with a unity.The authors defined that R has a generalized triangular matrix representa- tion if it is ring isomorphic to a generalized triangular matrix ring where , and the matrices obey the usual rules for matrix addition and multiplication.Let R ( ) ℓ and R r ( ) denote the sets of left and right semicentral idempotents of R. For some idempotent, ∈ e R it follows that = eRe e 0, ( ) { } ℓ if and only if = eRe e 0, r ( ) { }.When this property for e is satisfied, e is called semicentral reduced.A ring is called semicentral reduced if 1 is a semicentral reduced idempotent.
If each R i in matrix ( 1) is semicenral reduced, then the ring R has a complete generalized triangular matrix representation. An From the main result of the paper (Theorem 2.9), it follows that the ring R has a complete generalized triangular matrix representation if and only if R has a complete set of left triangulating idempotents.
In 2000, Heatherly and Tucci [54] presented properties of central and semicentral idempotents of a ring.Birkenmeier et al. [55] provide a survey of results on generalized triangular matrix algebras and semicentral reduced algebras.
New results have been developed for endomorphism algebras of modules and semicentral reduced algebras.One of them is given as follows: Theorem 3.1.For any positive integer n, R is semicentral reduced if and only if M R n ( ) is semicentral reduced.
The authors show that the semicentral reduced rings exhibit behavior similar to that of prime rings.For that reason, they develop various criteria which ensure that a semicentral reduced ring is prime.
In 2003, Birkenmeier and Park [56] described the semicentral idempotents of various ring extensions of a ground ring R in terms of the semicentral idempotents of R.They proved that if R is quasi-Baer then R has a triangulating dimension n if and only if R has exactly n minimal prime ideals.Most of the results are applied to determine complete generalized triangular matrix representations for various ring extensions of a ground ring R.
In 2020, Ánh, et al. [57] established a new way to unify and expand the classical theory of semiperfect rings to a much larger class of rings.The authors constructed the so-called Peirce trivial idempotents that generalize the notion of semicentral idempotents in the structure of 2-by-2 generalized triangular matrix rings.
For a ring R, the basic definitions are as follows: For any natural n, it follows that n-Peirce rings are generalizations of rings with a complete set of triangulating idempotents [52].
The first main result is as follows: Theorem 3.2.Let R be an n-Peirce ring and ∈ e R an arbitrary Peirce trivial idempotent.Then eRe is a k-Peirce ring for some ≤ k n and − Another significant outcome reveals that the authors have effectively formulated a structural theory for Peirce rings, mirroring the framework established by Bass for semiperfect rings.
Finally, Anh et al. [57], following Jacobson [58], construct the so-called trivial idempotents relative to certain radicals, like J-trivial and B-trivial idempotents, where J and B are Jacobson and prime (Baer) radical.

Derivations of matrix rings. Links to Amitsur's theorem
In 2022, Vladeva [59] offers a description of the R-derivations of R UTM n ( ), the ring of upper triangular matrices over an associative ring R with an identity.
A derivation The author considered the matrices , where ≤ ≤ k n 1 , and proved that k ℓ are left semicentral idempotents of the ring R UTM n ( ) (the matrix e ij has i j , ( )-entry 1 and rest zero is called a matrix unit).The inner derivations d k ℓ defined by 1 , for any matrix ∈ A R UTM n ( ) are idempotents and linearly independent in the additive group of R-derivations of the ring UTM R ( ).The author preferred to work with the derivations δ i such that = δ A e A , form a basis of the R-module .The main result of the paper follows that all derivations of R UTM n ( ) belong to .
Then there are the matrices , are the basic derivations.
, it appears that the result similar to the aforementioned theorem, when on the right side of the equality d k ℓ appears, will be true.Building upon Vladeva's proposal [59] to represent an arbitrary derivation using derivations generated by left semicentral idempotents in the case = n 3.

33
. Then there are the matrices Proof.It easily follows that By using the proof of Theorem 3.4 (Theorem 3.1 of [59]), we have , where ≤ ≤ ≤ p q 1 3. Then (as in the study by Vladeva [59]) we obtain that . Now for . Hence the result follows.□ In 2023, Vladeva [60] investigated the class of endomorphisms α of a ring R UTM n ( ) of upper triangular × n n matrices over an associative ring R with a unity.
Let us compare the two results: Proposition 2.5 in [59] and Proposition 2.5 in [60].In the first one, if ℓ is a left semicentral idempotent and r is a right semicentral idempotent of an arbitrary ring R and, moreover, x R define a derivation of R. From the second proposition without the restriction + = r 1 ℓ , it follows that = x rx Φ( ) ℓ for any ∈ x R define an endomorphism of R. Thus, we may conclude that the endomorphisms of an arbitrary ring, generated by left and right semicentral idempotents, are way more (in general) than the derivations.
For any left semicentral idempotent , is a commutative semigroup with an identity.The first main result of the article is as follows: ( ) or = α e 0 ii ( ) for all = i n 1,…, .All endomorphisms that belong to R n ( ) are regular (0,1)- endomorphisms.The second important result is presented as follows: Theorem 3.6.The class of regular (0,1)-endomorphisms is R n ( ).
We will use the following definition by Vladeva [64] An algebra = + S S, ,. ( ) with two binary operations + and ⋅ on S, is called a semiring if: for any ∈ a S.An element a of a semiring S is called additively idempotent if + = a a a.A semiring S is named additively idempotent if each of its elements is additively idempotent.Additively idempotent semirings are proper semirings, i.e., they are not rings.
Golan [61] comments on the following feature: "On one hand, semirings are abstract mathematical structures and their study is part of abstract algebra -arising from the work of Dedekind, Macaulay, Krull, and others ... On the other, the modern interest in semirings arises primarily from fields of applied mathematics...".
In addition, it is worth noting that semirings have significant connections to applied mathematics, linguistics, theoretical physics, cryptography, and various other scientific disciplines.

Applications of additively idempotent semirings
The majority of applications involving semirings pertain to additively idempotent semirings.

Automata theory and linguistics
Schützenberger [65] first acknowledged the significance of semirings in automata theory in 1961 when he formulated the theory of weighted automata and rational power series.Weighted finite automata hold both theoretical and practical importance in computer science, playing a pivotal role in the structural analysis of recognizable languages.Furthermore, they find practical applications in fields such as speech recognition and image compression, as highlighted in the previous studies [66,67].

Logic and theoretical computer science
In 1969, Hoare [68] introduced a formal system, known as Hoare logic, to investigate specification and verification of computer programs.Recently, modal operators for idempotent semirings have been introduced to model properties of programs and transition systems more conveniently and to link algebraic formalisms with traditional approaches such as dynamic and temporal logics [69].Interpretations of logical formulas over additively idempotent semirings, excluding the Boolean semiring, find applications in various areas of computer science.
Many valued algebras were introduced by Chang [70] as the algebraic counterpart for the infinite valued logic of Łukasiewicz.Recently, Di Nola and Russo [71,72], using MV-algebras that are additively idempotent semirings (in fact, MV-algebras are inclines), have obtained new results of MV-algebras.

Optimizations and max-plus algebras
Max-algebra has been studied in research papers and books from the early 1960s.In 1960, Cuninghame-Green [73] produced the first paper of the topic followed by numerous other articles that were summarized in a lecture notes volume [74] in 1979.Max-algebra is the analog to linear algebra, developed over an additively idempotent semifield = ∪ −∞ max { } (with operations ⊕ = x y x y max , { } and ⊙ = + x y x y) [8,[75][76][77].The max-plus-based methods described in the monograph by McEneaney [78] are oriented towards solving a Hamilton-Jacobi partial differential equation.It covers as an important fact that the semigroup associated with the nonlinear Hamilton-Jacobi partial differential equation is a linear max-plus operator.

Tropical geometry
Tropical geometry can be thought of as algebraic geometry over the tropical semiring, a piecewise linear version of algebraic geometry, which replaces a variety by its combinatorial shadow.The foremost workers in this area are Mikhalkin [79,80], Itenberg et al. [81], and Sturmfels et al. [82,83].Some of the basic concepts for amoebas of algebraic varieties and their geometric properties are discussed in by Mikhalkin and Theobald [84,85].Very recent investigations on these topics have come from Maclagan and Rincon [86] and Ito [87].

Idempotent analysis
The superposition principle (in quantum mechanics) means that the Schrödinger equation is linear.Similarly, in idempotent analysis, the superposition principle means that some important and basic problems and equations (optimization problems, the Bellman equation and its versions and generalizations, the Hamilton-Jacobi equation, etc.), which are nonlinear in the usual sense, can be treated as linear over appropriate idempotent semirings, see studies by Maslov [88] and Maslov and Sambourskiĭ [89].

Petri nets
Modern technology has created dynamic systems that are not easily described by differential equations.The state of such dynamic systems changes only at discrete instants of time instead of continuously, and they are called discrete event dynamic systems.Timed Petri nets are one of the best studied and most widely known models of discrete event dynamic systems.A Petri net is called an event graph, if all arcs have the weight 1 and each place has exactly one input and one output transition.The fact that Petri nets are connected with additively idempotent semirings has been well known since 1992 [90].Recently, the previous studies [91,92] have initiated an algebraic study of Petri nets.

Cryptography
Modern cryptography is mostly public-key cryptography.One-way trapdoor functions are essential to the study of this subject.A one-way trapdoor function is a one-way function f from a set X to a set Y with the additional property, the trapdoor, and it becomes feasible to find for any ∈ y f Im( ), an ∈ x X such that = f x y ( ) . Recently, we have noted the investigations of Grigoriev and Shpilrain [93,94] and the monograph of Roman'kov [95].We would also like to note the Bulgarian contribution to these studies [96][97][98][99][100].

The research of Rowen and the mathematical community around him
It is impossible to review all the results of Rowen and the mathematical community around him, due to the large number of articles and their high quality.Perhaps if we were to write ten (or more) surveys called "Rowen: Tropical Algebra," "Rowen: Supertropical Semirings," "Rowen: Hyperfields," "Rowen: The Negation Map," and so on, we would make a small step toward doing this review.

Derivations of polynomial Amitsur's idea and semicentral idempotents
When it comes to polynomial semirings, we refer to Golan [61].It is well known that the polynomial algebra over an additively idempotent semiring does not satisfy unique factorization.For example, ) are two different factorizations of the polynomial . Very recently, Baily et al. [130], Dong [131], Akian et al. [132] have explored polynomial semirings.In 2000, Thierrin [133] first considered derivations of semirings.He proved that the semiring of languages over some alphabet forms an additively idempotent semiring.
Why are derivations so important for semirings?
Let S be a semiring and Hence, S 2 ( ) is a semiring with derivation δ.Note that the semiring S need not be additively idempotent.
Since S may be identified with subsemiring of S 2 ( ) consisting of matrices of the form ⎛ ⎝ ⎞ ⎠ a a 0 0 , it follows that every semiring can be embedded in a semiring with nontrivial derivation.In 2020, Vladeva [134] ) is a derivation.This derivation referred to as a generalized inner derivation.The key result, where the author proved, that every derivation is a sum of an S-derivation and a generalized inner derivation, is actually Amitsur's idea: In 2020, Vladeva [136] defined a multiplication in a noncommutative additively idempotent semiring of polynomials S x [ ] by the rule = + xa ax δ a ( ), where ∈ a δ a S , ( ) and δ is a derivation of S. It is worth noting that when considering examples of derivations of S she examined the multiplications by the so-called left (right) Ore elements of S, which are essentially left (right) semicentral idempotents, as we show in the last section.
For the aforementioned derivation δ the author constructs a map A derivation , is a derivation of S x [ ]. Vladeva [134] showed that ∂ defined by The main result is similar to Amitsur's theorem:

Derivations of matrix semirings. Amitsur's idea and semicentral idempotents
When it comes to the history of matrix semirings, we refer to the survey paper of Gondran and Minoux [137].
In 1999, Minoux [138] generalized a well known theorem of graph theory using matrices over semiring.
In 2020, Tan [140] obtained a condition for an idempotent matrix over a commutative semiring to be diagonalizable.
In 2022, Vladeva [141]  , where Moreover, is an S-semimodule and the derivations are the elements of the basis of .
In the central conclusion of the paper, we obtain the Amitsur's idea: ( ) ( ), where S is an additively idempotent semiring, is a linear combination of elements of the basis of the S-semimodule with coefficients from S.
In 2021, Vladeva [142] studied the semiring M S n ( ) of × n n matrices over an additively idempotent semiring S. In the next lemma, the author represented the derivatives of the matrix units under an arbitrary S-derivation.Similar studies on semirings of endomorphisms can be found in the study by Vladeva [143].
4.5 Derivations of endomorphism semirings.Derivations of some classes of additively idempotent semirings.Amitsur's idea and semicentral idempotents In a set of n elements, say }. Thus, we define a chain with n elements.For a finite chain , the endomorphisms form a semiring with regard to the addition and multiplication defined by: This semiring is called the endomorphism semiring of n as is denoted by n .
In 2016, Vladeva [146] noted that by virtue of Theorem 2.2 of Kim et al. [147], it follows that any finite additively idempotent semiring can be represented as the endomorphism semiring of a finite chain.
. The elements of this are endomorphisms − a b k n k , where = k n 0,…, .An impor- tant equality for studying the derivations (in [146] of a special type) is the representation Therefore, the elements of subsimplices a a , , where ≤ ≤ − m k 1 1 , form an additive base of the simplex.
In 2018, Vladeva [148], using the aforementioned equality, proved that the of the considered simplex − σ a a a , , …, onto the subsimplices of an arbitrary type are derivations.In 2020, Vladeva [149] considered the maps The author constructed a semiring with this property is if and only if α 0 is an idempotent endomorphism.In the final result of this article, it is proved that the number of the semirings ∫α d m 0 1, , where ≤ ≤ − m n 1 1is the m 2 th Fibonacci number.With regards to the various types of the endomorphism semirings of an infinite chain, the reader is referred to the monograph Vladeva [150].
The subsemiring of S, generated by ∪ S S ( ) ( ), is denoted by S ( ).
The following three definitions are basic for much of what follows: (1) A semiring S is called -semiring if = S S ( ) .( 2) Let S and S 0 be additively idempotent semirings with a zero and an identity.Let S be a noncommutative semiring and an S 0 -semimodule, with S 0 a commutative semiring and = αs sα for any ∈ α S 0 and ∈ s S. Then S is called an S 0 -semialgebra.The semiring S from the aforementioned theorems is called an -matrix semiring over the semiring S 0 .The core discovery by Vladeva [152] refers to the derivations of an -matrix semiring S. If k ℓ is a left semicentral idempotent of S, then the map → δ S S : The first result following the Amitsur's idea is the next theorem.
Theorem 4.13.Let S be an -matrix semiring which is an S 0 -semialgebra with an e-basis e ij { }, where = i j n , 1,…, and ≤ i j.Let D be an arbitrary derivation of S. Then there is a derivation ( ) ( ), where = i j n , 1,…, and ≤ i j.
By replacing the arbitrary commutative semiring S 0 with the Boolean semiring the author obtained a stronger result: Theorem 4.14.Let S be an -matrix semiring which is a -semialgebra with an e-basis e ij { }, = i j n , 1,…, , ≤ i j.All arbitrary derivations of the semiring S are elements of the additively idempotent semiring .The set of nilpotent elements of S is an ideal, which is closed under all of these derivations.
In conclusion, our objective is to scrutinize our paper within the broader context of the development of additively idempotent semirings theory.
The interest in the applications of additively idempotent semirings is of great importance, since we do not know what the foundations of the theory of these semirings are.Therefore, you can refer to the books of Głazek [5] and Golan [61] to gain information on their applications from the last century.You can find details on more recent papers in Subsection 4.1; however, it is important to acknowledge that over the last two decades, more than a thousand papers with similar themes have been published.In Subsection 4.2, we have specifically highlighted only a selection of papers by Rowen and the mathematical community associated with him, focusing on those that contribute to the future trajectory of additively idempotent semirings theory.In the next three subsections, we present the derivations as an important tool for studying the various additively idempotent semirings.
Perhaps (as in the beginning of the development of differential algebra), the appearance of a book, as Kaplansky's little book [153], will be a key moment for the future of the theory of additively idempotent semirings.Following the ideas of Kaplansky [153] and Kolchin's monograph [154], many articles on differential algebra appeared [2].
That is why we expect bright horizons for the theory of additively idempotent semirings.

Theorem 2 . 1 .
Let R be a ring with unity and let → a derivation.Then there exists a derivation→ δ R R : and a matrix ∈ A T R n ( ) such that = + d δ d A ͠ .
) be an arbitrary R-derivation of the ring R UTM 3 ( ) ) distributive laws hold ⋅ + = ⋅ + ⋅ semiring S is called commutative if ⋅ = ⋅ a b b a for any ∈ a b S, .In some of the considered semirings S are assumed to exists a zero element ∈ S 0

Theorem 4 . 2 .
Let S be a commutative additively idempotent semiring.For each derivation → D S x S x : [ ] [ ], there exists a generalized inner derivation Δ D and a polynomial

Theorem 4 . 3 .
is the center of S and Co S δ ( ) is the subsemiring of S, consisting the constants δ, then = a nontrivial subsemiring of S. The author proved that = derivations of S x [ ] and then state the result as follows: Each element of the commutative additively idempotent semiring S d Γx ( )[ ] is a derivation of S x [ ].

Theorem 4 . 4 .
Let S be an additively idempotent semiring.For an arbitrary δ -derivation → there exists a derivation Δ D such that =

11 ,
explored the semiring S UTM n ( ) of upper triangular matrices over an additively idempotent semiring S. By using the sums = the author constructed a derivation δ k such that = δ n ( ).But D k and D m are just the left and right semicentral idempotents, respectively, con- sidered in[59] (see Subsection 3.2).The set of derivations δ k and d m is an additively idempotent semiring such that for the products δ d k m that the author proved Theorem 4.5.Let ∈ δ d , k m

Theorem 4 . 9 .( 4 )
Let S be an arbitrary additively idempotent semiring, LO S ( ), the subsemiring of the left semicen- tral idempotents of S, and RO S ( ) be the subsemiring of the right semicentral idempotents of S. Let M S n ( ) be the semiring of × n n matrices over S and → ), the elements of the matrix D A ( ) are d g a ij ( ) ℓ , where d a ij ℓ are the derivations, generated by a ij and ∈ g S is a generator of D. ), the elements of the matrix D A ( ) are d g a r ij ( ), where d a r ij are the derivations, generated by a ij and ∈ g S is a generator of D. (3) If D is a left derivation in M S n ( ), then for an arbitrary = ∈ a generator of D, and therefore, D is a hereditary derivation.If D is a right derivation in M S n ( ), then for an arbitrary = ∈ where ∈ g RO S ( ) is a generator of D, and hence, D is a hereditary derivation.

1
is a subsemiring of n of order the n-th Catalan number.Moreover, the set of nilpotent endomorphisms n is an ideal of n of order the − n 1 ( )-th Catalan number.Further, in the article, new derivations d m , ℓ are constructed and, for a fixed endomorphism α 0 , the set of endomorphisms α such that =

( 3 ) 1 , where 1 S
Let S be an S 0 -semialgebra and = basis of S with the following properties: is an identity of S. called an e-basis.The important results are as follows.Theorem 4.10.Let S be an S 0 -semialgebra and = an e-basis of S. Then S is isomorphic to a matrix subsemiring of M S n 0 ( ).

Theorem 4 . 11 .
Let S be an S 0 -semialgebra with an e-basis =

1 .
, where ∈ a S is a derivation of S. Similarly, if r m is a right semicentral idempotent, then the map → is a derivation of S. Let be the semiring generated by the set of all derivations δ k and d m .For the products δ d k m , we have the following result: Theorem 4.12.Let S be an -matrix semiring.Let ∈ The map δ d k m is a derivation if and only if + ≥ k m n.
Representation of derivations of matrix ringsThroughout the discussion, unless otherwise mentioned, R denotes an associative (not necessarily commu- tative) ring.Recall that a derivation of R is an additive map a ( ) [ ] for any ∈ x R is a derivation called inner derivation of R determined by a.
Peirce ring if 0 and 1 are the only Peirce trivial idempotents of R. For a natural number > n 1, a ring R is called an n-Peirce ring if there is a Peirce trivial idempotent ∈ e R such that eRe is an m-Peirce ring for some m, ≤ < is called inner Peirce trivial if − = eR e Re 1 0 ( ) and is called outer Peirce trivial if − e 1 is an inner Peirce trivial.An idempotent e is Peirce trivial if it is both inner and outer Peirce trivial.A ring R is called a 1- Following Jacobson[135, p. 530], we can call a derivation an S-derivation if it an S-linear map.Therefore, the following important conclusion can be drawn: d [142] 4.7.Let S be an arbitrary (not necessarily commutative) additively idempotent semiring and M S where g is the generator of D. The generalization of the previous theorem for noncommutative semiring S is the last theorem in Vladeva[142]: