The strong nil-cleanness of semigroup rings

Abstract In this paper, we study the strong nil-cleanness of certain classes of semigroup rings. For a completely 0-simple semigroup M = ℳ 0 ( G ; I , Λ ; P ) M={ {\mathcal M} }^{0}(G;I,\text{Λ};P) , we show that the contracted semigroup ring R 0 [ M ] {R}_{0}{[}M] is strongly nil-clean if and only if either | I | = 1 |I|=1 or | Λ | = 1 |\text{Λ}|=1 , and R [ G ] R{[}G] is strongly nil-clean; as a corollary, we characterize the strong nil-cleanness of locally inverse semigroup rings. Moreover, let S = [ Y ; S α , φ α , β ] S={[}Y;{S}_{\alpha },{\varphi }_{\alpha ,\beta }] be a strong semilattice of semigroups, then we prove that R [ S ] R{[}S] is strongly nil-clean if and only if R [ S α ] R{[}{S}_{\alpha }] is strongly nil-clean for each α ∈ Y \alpha \in Y .


Introduction
Diesl [1] introduced the concept of nil-clean and strongly nil-clean rings and asked the question when a matrix ring is (strongly) nil-clean. As we have observed, it is difficult to characterize the arbitrary ring R such that ( ) R n is nil-clean [2]. It is also known that the ring of all × 2 2-matrices over any commutative local ring is not strongly nil-clean [3]. On the other hand, the strong nil-cleanness of some generalizations of matrix rings has been considered. By [1], if R is a commutative ring with identity, then the formal block matrix ring is strongly nil-clean if and only if A B , are strongly nil-clean. Moreover, the strong nilcleanness of Morita contexts, formal matrix rings and generalized matrix rings has also been studied [2]. Noting that semigroup rings are also an extension of group rings, the strong nil-cleanness of certain kinds of group rings has been studied in [2,4,5].
However  [6]. Note that the term "non-unital ring" or "general ring" refers to a ring that does not necessarily have an identity. It is known that Nicholson extended many of the results on the (strong) cleanness of rings with identity to non-unital rings [7], and that in [1,2], the (strong) nil-cleanness of non-unital rings has been studied.
The paper is organized as follows. In Section 2, some preliminaries are given. In Section 3, we show that the strong nil-cleanness of the contracted semigroup ring can be characterized by | | I , | | Λ and the strong nil-cleanness of [ ] R G . As an application, we determine when the contracted semigroup ring of a locally inverse semigroup with idempotents set locally finite is strongly nil-clean. In Section 4, we consider the strong nil-cleanness of certain classes of supplementary semilattice sums of rings, and the relationship between the strong nil-cleanness of contracted semigroup rings and semigroup rings.
Throughout this paper, a ring always means an associate non-unital (or general) ring, and we always assume that R is a ring with identity (not necessarily commutative).

Preliminaries
In this section, the notations and definitions on semigroups and semigroup rings which will be used in the sequel are provided, see [6,8,9].
Unless otherwise stated, a semigroup S is always assumed to have a zero element (denoted by θ or θ S ). Denote by S 1 the semigroup obtained from S by adding an identity if S has no identity, otherwise, let = S S 1 . Green's equivalence relations play an important role in the theory of semigroups, which were introduced by Green (1951): for ∈ a b S , , . A semigroup is said to be regular if every -class and every -class of it contains an idempotent. An inverse semigroup is a regular semigroup with commutative idempotents. Let ( ) = { ∈ | = } E S e S e e 2 be the set of idempotents of S. We call a regular semigroup S a locally inverse semigroup if eSe is an inverse subsemigroup of S for each ∈ ( ) e E S . It is clear that an inverse semigroup is locally inverse. We say the idempotent set ( ) E S of a locally inverse semigroup S is locally finite if ( ) E eSe is finite for each ∈ ( ) e E S . We call a semigroup S a completely 0-simple semigroup if S is 0-simple and all its non-zero idempotents are primitive. Let G be a group, I and Λ be two non-empty sets and let = ( ) P p λi be a × I Λ -matrix with entries in G 0 (= ∪ { }) G 0 , and suppose that P is regular, in the sense that no row or column of P consists entirely of zeros. Let = ( × × ) ∪ { } M I G Λ 0 and define a multiplication on M by Then M is a completely 0-simple semigroup, denoted by ( ) G I P ; ,Λ; 0 . Note that the zero element of a completely 0-simple semigroup is denoted by 0 instead of θ. Conversely, every completely 0-simple semigroup is isomorphic to one constructed in this way.
In particular, if P is with entries in G, then the set ( × × ) I G Λ forms a subsemigroup of ( ) G I P ; ,Λ; 0 , we denote this subsemigroup by ( ) G I P ; ,Λ; , and call it a completely simple semigroup.
Recall the definition of a Munn ring. Let T be a ring, I, Λ be two non-empty sets. It will be necessary to describe a × I Λ-matrix over T in terms of its entries. In this case, we use such notations as Note that the aforementioned definitions and results on Jacobson radicals are valid with "left" replaced by "right." 3 Strong nil-cleanness of completely 0-simple semigroup rings The first part of this section presents some elementary definitions and results on strongly nil-clean rings. The remainder of this section is concerned with the strong nil-cleanness of the contracted semigroup rings of both completely 0-simple semigroups and locally inverse semigroups.
Let U be a ring. Following Diesl [1], an element ∈ u U is said to be nil-clean if there is an idempotent ∈ e U and a nilpotent ∈ b U such that = + u e b. The element u is further said to be strongly nil-clean if such an idempotent and a nilpotent can be chosen to satisfy the equality = be eb. The ring U is called a nil-clean (respectively, strongly nil-clean) ring if each element in it is nil-clean (respectively, strongly nil-clean).
We establish a basic result which will be used later.
The following lemma provides a very useful characterization of strongly nil-clean rings, in terms of Jacobson radicals. A ring is said to be boolean if every element of the ring is an idempotent.
By Lemma 3.2, it will be helpful to know the Jacobson radicals of contracted completely 0-simple semigroup rings.
Diesl proved the inheritance of the strong nil-cleanness by corner subrings of a strong nil-cleanly ring with identity from the ring with identity. This remains true for non-unital rings by applying similar argument in the proof of [1, Proposition 3.25].
Lemma 3.4. Let U be a ring and f be any idempotent in U. If U is strongly nil-clean, then the ring fUf is also strongly nil-clean.
Proof. Since Uf is a left ideal of U and fUf is a right ideal of Uf, we apply Lemma 3.1 to obtain first that Uf is a strongly nil-clean ring, and then that fUf is a strongly nil-clean ring. The lemma is proved. □ Later, Lemma 3.4 is used to give a necessary condition for a contracted completely 0-simple semigroup ring to be strongly nil-clean. is not strongly nil-clean.
Proof. By hypothesis, the completely 0-simple semigroup M is of the form Since M is a regular semigroup, by Lemma 2.1, the sandwich matrix P must be one of the matrices in the set { ′ ′ } N N N N N Δ, , , , , 3 , up to isomorphic, where and where e is the identity of the group G and ∈ g G. We claim that the contracted semigroup ring [ ] R M 0 is not strongly nil-clean, no matter which of the above normalized matrix P is chosen. There are five cases to be considered. For convenience, let denote the set consisting of all the fields of characteristic 2.
is strongly nil-clean by assumption, [ ] R G is strongly nil-clean by Lemma 3.4. Then it follows from Lemma 3.

that ( [ ])
J R G is nil. Since g is an invertible element in G, we deduce that ( − ) cg e c is a nilpotent element. Since c is arbitrary, we can take = − − c g 1   is not strongly nil-clean. This may be thought of as an extension of the result in [3]. By applying Lemmas 3.1 and 3.5, a necessary condition for a contracted completely 0-simple semigroup ring to be strongly nil-clean is obtained.    Therefore, x is a nilpotent element when ∈ ( [ ]) x J R M 0 , which yields that  We have obtained all the preliminaries needed to prove the following principal theorem in this section. is strongly nil-clean, it follows from the proof of Theorem 3.8 that all entries of P must be non-zero, and hence M must be of the form ( )∪{ } G I P ; ,Λ; 0 . In the following, we provide two corollaries of Theorem 3.8. is strongly nil-clean for each ∈ ( / ) * α S . Furthermore, for each ∈ ( / ) * α S , the maximal subgroup of D α 0 is isomorphic to the maximal subgroup in D α , and the number of -classes (resp., -classes) in D α 0 is equal to the number of -classes (resp., -classes) in D α , see [13]. The result then follows from Theorem 3.8. □

Strong nil-cleanness and strong semilattice of semigroups
We recall the definition of strong semilattice of semigroups. Let Y be a semilattice with natural partial order ≤ and S α be a family of semigroups indexed by Y. We say a semigroup S is a strong semilattice of semigroups S α ( ∈ ) α Y if S is the disjoint union of the semigroups S α , and for all > α β in Y there exists a homomorphism → φ S S : α β α β , satisfying the following conditions: , ; (iii) If ∈ a S α and ∈ b S β , then the multiplication in S is given by . Then . It follows from the maximality of β that + + ( ) = zx y y zx 0 is a ring isomorphism from A to * A .
Note that Lemma 4.3 is true for the more general case when Y is taken to be a pseudofinite semilattice [14]. , as required. □ The following result provides us an equivalent characterization for a non-unital ring to be strongly nilclean, which plays an important role in the below proof.  We end this paper with a brief discussion of the relationship between the strong nil-cleanness of semigroup rings and the strong nil-cleanness of contracted semigroup rings. Let N be a semigroup. is not strongly nil-clean. By contrary, suppose that = + ( − ) a θ ka kθ 2 for some ∈ k K, then k must be equal to 1 K , whence = a a 2 , which is a contradiction.
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