Generalized Munn rings

Junying Guo; Xiaojiang Guo

doi:10.1515/math-2022-0487

Open Access Published by De Gruyter Open Access October 6, 2022

Generalized Munn rings

Junying Guo and Xiaojiang Guo

From the journal Open Mathematics

https://doi.org/10.1515/math-2022-0487

Abstract

Generalized Munn rings exist extensively in the theory of rings. The aim of this note is to answer when a generalized Munn ring is primitive (semiprimitive, semiprime and prime, respectively). Sufficient and necessary conditions are obtained for a generalized Munn ring with a regular sandwich matrix to be primitive (semiprimitive, semiprime and prime, respectively). Also, we obtain sufficient and necessary conditions for a Munn ring over principal ideal domains to be prime (semiprime, respectively). Our results can be regarded as the generalizations of the famous result in the theory of rings that for a ring R , R is primitive (semiprimitive and semiprime, respectively) if and only if so is M n ( R ) . As applications of our results, we consider the primeness and the primitivity of generalized matrix rings and generalized path algebras. In particular, it is proved that a path algebra is a semiprime if and only if it is semiprimitive.

Keywords: generalized Munn ring; generalized path algebra; semiprimitive ring; semiprime ring; generalized matrix ring

MSC 2010: 16S36; 15A30; 05C25

1 Introduction

Throughout this note, we shall use the standard notions and notations, and each of the considered rings is associative but has possibly no identity.

The class of generalized matrix rings has been extensively studied. Examples of generalized matrix rings include piecewise domains (see [1]), incidence algebras of directed graphs (see, [2,3]), structural matrix rings (see [4] and subsequent papers), endomorphism rings, and Morita context rings. Sands [5] observed that if [ S , V , W , T ] is a Morita context, then

S V W T

is a ring. These Morita context rings are precisely generalized matrix rings with idempotent sets E such that ∣ E ∣ = 2 , and they have been widely studied. In particular, we note Amitsur’s paper [6], the survey paper [7], McConnell and Robson’s treatment [8] and Müller’s computation of the maximal quotient ring [9]. Indeed, cellular algebras, affine cellular algebras and standardly based algebras there exist some “local” structures of generalized Munn algebras (for example, see [10,11,12]).

Brown [13] considered generalized matrix algebras of finite dimension over a field of characteristic 0. He proved that such a generalized matrix algebra is either simple or nonsemisimple and simple modulo its radical, and it is simple if and only if it possesses an identity. Sands [5] gave the prime radical of generalized matrix rings with a finite idempotent set. Zhang [14] considered the prime radical of the general case. In 1989, Wauters and Jespers [15] determined when a generalized matrix ring with a finite idempotent set is semiprime. Classical quotient rings of generalized matrix rings with finite idempotent sets had been attracting due attention. There are a series of papers on this field (see [16,17,18]).

Indeed, any generalized matrix ring can be viewed as a subring of some generalized Munn ring. It is natural to research generalized Munn rings. Li [19] considered the regularity of Munn rings. The main aim of this study is to answer when a generalized Munn ring is semiprimitive (semiprime and prime, respectively). A sufficient and necessary condition is established for a generalized Munn ring with a regular sandwich matrix to be primitive (semiprimitive, semiprime and prime, respectively) (Theorems 2.5 and 2.7). Moreover, we answer when a Munn ring over a principal ideal domain is semiprime (prime, respectively) (Proposition 2.11). In Section 3, we determine when a generalized matrix ring is primitive (semiprimitive, semiprime and prime, respectively) (Theorems 3.2 and 3.3). Finally, we consider the primeness and the primitivity of generalized path algebras. It is proved that for a quiver Q and a field K , the path algebra K ( Q ) is semiprime if and only if the path-connected quiver of Q ¯ P C is the disjoint union of complete quivers; if and only if K ( Q ) is semiprimitive (Theorem 4.9). And, K ( Q ) is prime if and only if Q ¯ P C is a complete quiver (Theorem 4.10).

2 Generalized Munn rings

The aim of this section is to consider the primeness and the primitivity of generalized Munn rings.

2.1 Regular sandwich matrices

To begin with, we give the notion of regular matrices.

Definition 2.1

Let A be a ring and X = ( x i m ) be a generalized I × M matrix over A . A nonzero entry x i m of X is called a unit entry of X if there exists a nonzero idempotent e ∈ A such that x i m is a unit in e A e .

Notice that a group has exactly one idempotent, which is just the identity of the group. This means that in Definition 2.1, the idempotent e is indeed unique. So, the unique idempotent e in Definition 2.1 is denoted by x i m ◇ .

Definition 2.2

Let A be a ring. An I × M matrix X = ( x i m ) over A is said regular in A if the following conditions hold:

For any i ∈ I , there exists m ∈ M such that x i m is a unit entry of X .
For any n ∈ M , there exists j ∈ I such that x j n is a unit entry of X .
If x i 0 m 0 is a unit entry of X , then
1. x i 0 m 0 ◇ x i 0 m = x i 0 m for any m ∈ M ;
2. x i m 0 x i 0 m 0 ◇ = x i m 0 for any i ∈ I .

By definition, any m × m matrix without zero rows and zero columns are regular in the field C of complex numbers. Also, for a ring A with unity, any I × M matrix over A , in which each row and each column contains at least one unit of A , must be regular in A .

Let M , I be nonempty sets, and A an associative ring and Q = ( q m i ) a generalized M × I matrix over A . Consider the set M ( A , I , M ) consisting of all generalized I × M matrices over A with only finite nonzero entries, such an I × M matrix is usually said to be bounded. For C = ( a i m ) , D = ( b i m ) ∈ M ( A , I , M ) , define

C + D = ( e i m ) , where e i m = c i m + d i m for i ∈ I , m ∈ M ; C ∘ D = C Q D , where the product on the right side is the product of matrices; λ C = ( λ c i m ) for λ ∈ R .

By definition, a routine calculation shows that with these operations, M ( A , I , M ) is an associative ring.

Definition 2.3

The above ring M ( A , I , M ) is called a generalized Munn ring A with the sandwich matrix Q , in notation, M ( A , I , M ; Q ) .

If I is finite, then we identity it with the set { 1 , 2 , … , i } , where i is the cardinality of I , and we write M ( A , I , M ; Q ) as M ( A , i , M ; Q ) . Similarly, the notation M ( A , I , m ; Q ) is used if ∣ M ∣ = m < ∞ . Denote by M m , n ( A ) the set of all m × n matrices over A .

Recall from [20] that the generalized Munn ring M ( A , m , n ; Q ) is called the Munn m × n matrix ring over A with sandwich matrix Q . It is obvious that M n ( A ) is the Munn n × n matrix ring over A with sandwich matrix Δ , where Δ is the unit matrix; that is, the diagonal matrix each of whose entries in the diagonal positions is the unity of A .

Definition 2.4

Let T be a nonempty subset of M ( A , I , M ) . An M × I matrix X = ( x m i ) over A is said to be cancellable in T if for any nonzero element Y ∈ T , Y X and X Y are neither zero, where Y X and X Y are usual matrix products.

Evidently, for a ring A with identity 1, the I × I unit matrix Δ is cancellable in any subset of M ( A , I , I ) . And, any invertible n × n matrix must be cancellable in M n ( A ) , but not all of cancellable matrices in M n ( A ) are invertible in the matrix algebra.

Example. Let Z be the ring of integers. It is easy to check that the matrix

A = 2 0 0 4

is cancellable in the matrix ring M 2 ( Z ) . But A is not invertible in M 2 ( Z ) .

For convenience, we denote

( a ) i m : the generalized I × M matrix with a in the ( i , m ) position and 0 elsewhere; ( ℬ ) i m : the set { ( b ) i m : b ∈ ℬ } for ℬ ⊆ A ; ( a i m ) i ∈ J , m ∈ N : the generalized I × M matrix with a i m in the ( i , m ) position for i ∈ J , m ∈ N and 0 elsewhere. Especially, if N = { n } , we simply write m ∈ N as m = n , and the similar sign for the case : ∣ I ∣ = 1 . rad ( A ) : the Jacobson radical of the ring A .

We now arrive at our main results of this note.

Theorem 2.5

Let M = M ( A , I , M ; Q ) be a generalized Munn ring. If Q is regular in A , then M ( A , I , M ; Q ) is semiprime (semiprimitive, respectively) if and only if the following conditions are satisfied:

Q is cancellable in M ( A , I , M ) ;
for any unit entries q m i , q n j of Q , if x is a nonzero element of q m i ◇ A q n j ◇ , then q n j ◇ A q m i ◇ x ≠ 0 and x q n j ◇ A q m i ◇ ≠ 0 ;
for any unit entry q m i of Q , q m i ◇ A q m i ◇ is semiprime (semiprimitive, respectively).

Proof

Let q m i be a unit entry of Q and denote by q m i − 1 the inverse of q m i in q m i ◇ A q m i ◇ , obviously q m i ◇ = q m i q m i − 1 = q m i − 1 q m i . Then ( q m i − 1 ) i m is an idempotent of M , and

(2.1) ( q m i − 1 ) i m ∘ M ∘ ( q m i − 1 ) i m = ( q m i ◇ A q m i ◇ ) i m

since q m i is a unit in q m i ◇ A q m i ◇ . A routine calculation shows that the mapping

(2.2) ϕ : ( x ) i m ↦ x q m i is an isomorphism from ( q m i ◇ A q m i ◇ ) i m onto q m i ◇ A q m i ◇ .

(2.5.1) The proof for the semiprime case. If M is semiprime, then by (2.1), ( q m i ◇ A q m i ◇ ) i m is semiprime, so that by ϕ is an isomorphism, q m i ◇ A q m i ◇ is semiprime. It results (iii). To see (i), assume on the contrary that Q is not cancellable in M ( A , I , M ) , then there is a nonzero element X ∈ M such that X Q = 0 or Q X = 0 . Without loss of generality, let X Q = 0 , so that X ∘ U ∘ X = X Q U Q X = 0 for any U ∈ M , whence X ∘ M ∘ X = 0 , contrary to that M is semiprime. Thus, Q is cancellable in M ( A , I , M ) .

We next verify (ii). To the end, we assume contrariwise that there exists a nonzero element x ∈ q m i ◇ A q n j ◇ such that x q n j ◇ A q m i ◇ = 0 . Obviously, x q n j ◇ = x = q m i ◇ x , so that

( x ) i m ∘ M ∘ ( x ) i m ⊆ ( x A x ) i m = ( ( x q n j ◇ ) A ( q m i ◇ x ) ) i m = ( x ( q n j ◇ A q m i ◇ ) x ) i m = 0 ,

contrary to the hypothesis that M is semiprime. Therefore, x q n j ◇ A q m i ◇ ≠ 0 . Similarly, q n j ◇ A q m i ◇ x ≠ 0 . We have now proved that (ii) is valid.

For the converse, we contrariwise let w = ( w i m ) be a nonzero generalized I × M matrix in M such that w ∘ M ∘ w = 0 . Because Q is cancellable, Q w Q = ( u m i ) m ∈ M , i ∈ I ≠ 0 , and we assume that u m 0 i 0 ≠ 0 . If w Q = ( v i j ) i , j ∈ I , then

(2.3) u m 0 i 0 = ∑ j ∈ I q m 0 j v j i 0 .

When Q is regular, there is j 0 ∈ I such that q m 0 j 0 is a unit entry of Q . It follows that q m 0 j 0 ◇ q m 0 j = q m 0 j . Now by (2.3),

u m 0 i 0 = ∑ j ∈ I q m 0 j 0 ◇ q m 0 j v j i 0 = q m 0 j 0 ◇ ∑ j ∈ I q m 0 j v j i 0 = q m 0 j 0 ◇ u m 0 i 0 ,

and similarly, there exists a unit entry q n 0 i 0 of Q such that u m 0 i 0 q n 0 i 0 ◇ = u m 0 i 0 . Therefore, u m 0 i 0 ∈ q m 0 j 0 ◇ A q n 0 i 0 ◇ . Furthermore, by (ii), there is x ∈ q n 0 i 0 ◇ A q m 0 j 0 ◇ such that 0 ≠ u m 0 i 0 x ∈ q m 0 j 0 ◇ A q m 0 j 0 ◇ . Clearly,

(2.4) q m 0 j 0 ◇ u m 0 i 0 x = u m 0 i 0 x = ( u m 0 i 0 x ) q m 0 j 0 ◇ .

It follows that

(2.5) u ≔ ( q m 0 j 0 ◇ ) j 0 m 0 Q w Q ( x ) i 0 m 0 = ( q m 0 j 0 ◇ ) j 0 m 0 ( u m 0 i 0 ) m 0 i 0 ( x ) i 0 m 0 = ( q m 0 j 0 ◇ u m 0 i 0 x ) j 0 m 0 = ( u m 0 i 0 x ) j 0 m 0 = ( q m 0 j 0 ◇ ) j 0 m 0 ∘ w ∘ ( x ) i 0 m 0 ≠ 0 .

Now

( u m 0 i 0 x ⋅ q m 0 j 0 ◇ A q m 0 j 0 ◇ ⋅ u m 0 i 0 x ) j 0 m 0 = ( u m 0 i 0 x q m 0 j 0 ◇ ⋅ A ⋅ q m 0 j 0 ◇ u m 0 i 0 x ) j 0 m 0 = ( u m 0 i 0 x ) j 0 m 0 Q ( q m 0 j 0 − 1 A q m 0 j 0 − 1 ) j 0 m 0 Q ( u m 0 i 0 x ) j 0 m 0 ⊆ ( u m 0 i 0 x ) j 0 m 0 Q M Q ( u m 0 i 0 x ) j 0 m 0 = ( u m 0 i 0 x ) j 0 m 0 ∘ M ∘ ( u m 0 i 0 x ) j 0 m 0 = ( ( q m 0 j 0 ◇ ) j 0 m 0 ∘ w ∘ ( x ) j 0 m 0 ) ∘ M ∘ ( ( q m 0 j 0 ◇ ) j 0 m 0 ∘ w ∘ ( x ) j 0 m 0 ) ⊆ ( q m 0 j 0 ◇ ) j 0 m 0 ∘ ( w ∘ M ∘ w ) ∘ ( x ) j 0 m 0 = 0 ,

so that u m 0 i 0 x ⋅ q m 0 j 0 ◇ A q m 0 j 0 ◇ ⋅ u m 0 i 0 x = 0 . This is contrary to that q m 0 j 0 ◇ A q m 0 j 0 ◇ is semiprime. Consequently, M is semiprime.

(2.5.2) The proof for the semiprimive case. It is well known that for a semiprimitive algebra A and an idempotent e ∈ A , e A e is still semiprimitive. So, the same reason as in (2.5.1) shows that the “if” part is valid. With notations in (2.5.1), if w ∈ rad ( M ) \ { 0 } , then by (2.5), 0 ≠ ( u m 0 i 0 x ) j 0 m 0 ∈ rad ( M ) . Notice that ( q m 0 j 0 − 1 ) j 0 m 0 is an idempotent of M , we can obtain that

( u m 0 i 0 x ) j 0 m 0 = ( q m 0 j 0 ◇ u m 0 i 0 x q m 0 j 0 ◇ ) j 0 m 0 ( by (2.3) ) = ( q m 0 j 0 − 1 ) j 0 m 0 ∘ ( u m 0 i 0 x ) j 0 m 0 ∘ ( q m 0 j 0 − 1 ) j 0 m 0 ∈ ( q m 0 j 0 − 1 ) j 0 m 0 ∘ rad ( M ) ∘ ( q m 0 j 0 − 1 ) j 0 m 0 = rad ( ( q m 0 j 0 − 1 ) j 0 m 0 ∘ M ∘ ( q m 0 j 0 − 1 ) j 0 m 0 ) = rad ( ( q m 0 j 0 − 1 A q m 0 j 0 − 1 ) j 0 m 0 ) = rad ( ( q m 0 j 0 ◇ A q m 0 j 0 ◇ ) j 0 m 0 ) .

This means that ( q m 0 j 0 ◇ A q m 0 j 0 ◇ ) j 0 m 0 is not semiprimitive. Now by (2.2), we observe that q m 0 j 0 ◇ A q m 0 j 0 ◇ is not semiprimitive. This is contrary to the hypothesis. It results the “only if” part.□

Remark 2.6

Let us turn back to the proof of Theorem 2.5. In (2.5.1), the proof of Condition (i) in the direct part has indeed proved that if M ( A , I , M ; Q ) is semiprime, then Q is cancellable in M ( A , I , M ) .

Theorem 2.7

Let M = M ( A , I , M ; Q ) be a generalized Munn ring. If Q is regular in A , then M ( A , I , M ; Q ) is prime (primitive, respectively) if and only if the following conditions are satisfied:

Q is cancellable in M ( A , I , M ) ;
for any unit entries q m i , q n j , q r k , q s l of Q , if x and y are nonzero elements of q m i ◇ A q n j ◇ and q r k ◇ A q s l ◇ , respectively, then x q n j ◇ A q r k ◇ y ≠ 0 ;
for any unit entry q m i of Q , q m i ◇ A q m i ◇ is prime (primitive, respectively).

Proof

(2.7.1) The proof for the prime case. If M is prime, then by (2.1), ( q m i ◇ A q m i ◇ ) i m is prime, so that by ϕ is an isomorphism, q m i ◇ A q m i ◇ is prime. By Remark 2.6, Q is cancellable in M ( A , I , M ) .

We next verify (ii). We contrariwise let x ∈ q m i ◇ A q n j ◇ \ { 0 } , y ∈ q r k ◇ A q s l ◇ \ { 0 } such that x q n j ◇ A q r k ◇ y = 0 . Then, x q n j ◇ = x and q r k ◇ y = y . Moreover,

( x ) i m ∘ M ∘ ( y ) i m = ( x A y ) i m = ( x ⋅ q n j ◇ A q r k ◇ ⋅ y ) i m = ( 0 ) i m = 0 .

It is contrary to the hypothesis that M is prime. We have now proved the necessity.

To see the converse part, we assume conversely that there exist nonzero elements A , B ∈ M such that A ∘ M ∘ B = 0 . It is not difficult to see that Condition (ii) in Theorem 2.7 implies Condition (ii) in Theorem 2.5. Indeed, by (ii), for any x ∈ q m i ◇ A q n j ◇ , x q n j ◇ A q m i ◇ x ≠ 0 , so that x q n j ◇ A q n i ◇ ≠ 0 ; similarly, q n j ◇ A q m i ◇ x ≠ 0 , and it results immediately in Condition (ii) in Theorem 2.5. Now by (2.5.1) (precisely, see (2.4) and (2.5)), there are m , n ∈ M , j , k ∈ I and C 1 , C 2 , D 1 , D 2 ∈ M such that

q m j , q n k are unit entries of Q ; and
C 1 ∘ A ∘ D 1 = ( a ) j m and C 2 ∘ B ∘ D 2 = ( b ) k n , where a and b are nonzero elements in q m j ◇ A q m j ◇ and q n k ◇ A q n k ◇ , respectively.

Furthermore by (ii), we have u ∈ q m j ◇ A q n k ◇ such that a u b ≠ 0 . Obviously, q m j ◇ u = u = u q n k ◇ .

Compute

(2.6) ( a u b ) j n ∘ M ∘ ( a u b ) j n = ( a q m j ◇ u b ) j n ∘ M ∘ ( a u q n k ◇ b ) j n = ( a q m j q m j − 1 u b ) j n ∘ M ∘ ( a u q n k − 1 q n k b ) j n = ( a ) j m ∘ ( q m j − 1 u b ) j n ∘ M ∘ ( a u q n k − 1 ) j n ∘ ( b ) k n ⊆ ( a ) j m ∘ M ∘ ( b ) k n = C 1 ∘ A ∘ D 1 ∘ M ∘ C 2 ∘ B ∘ D 2 ⊆ C 1 ∘ A ∘ M ∘ B ∘ D 2 = 0 .

This shows that M is not a semiprime ring. But by Theorem 2.5, M is semiprime. It is a contradiction. Therefore, M is prime.

(2.7.2) The proof for the primitive case. By the well-known result (for example, see [21, Ex. 10, p. 339]): for any primitive algebra A and any idempotent e in A , e A e is still primitive, and since any primitive algebra is prime, a similar argument as in (2.5.1) can verify the “if” part. For the converse, if given conditions hold, then by (2.7.1), M is prime. The rest follows from a famous result of Lanahi et al. [22] showed that for a prime ring R , if e is a nonzero idempotent in R , then R is primitive if and only if e R e is primitive.□

Based on Theorems 2.5 and 2.7, we may prove the following proposition.

Proposition 2.8

Let M ( A , I , M ; Q ) be a generalized Munn ring. Assume that

A has a unity;
each row and each column of Q contains at least one unit of A .

Then M ( A , I , M ; Q ) is prime (semiprime, primtive and semiprimitive, respectively) if and only if the following conditions are satisfied:

Q is cancellable in M ( A , I , M ) ;
A is prime (semiprime, primtive and semiprimitive, respectively).

Proof

By definition, q m i ◇ is the unity of A for any unit entry q m i of Q satisfying Condition (2); in this case, q m i ◇ A q m i ◇ = A . Obviously, Q is regular in A .

Let q m i be an arbitrary unit entry of Q . By Condition (2), there is an entry q m i 0 of Q such that q m i 0 is a unit in A . But Q is regular in A , so q m i ◇ q m i 0 = q m i 0 , and it follows that q m i ◇ must be the unity of A . We have now proved that any unit entry of Q is a unit in A . This shows that Condition (ii) in Theorem 2.5 are satisfied and that Condition (ii) in Theorem 2.7 is satisfied whenever A is prime.

The rest follows immediately from Theorems 2.5 and 2.7.□

For a ring A with unity, denote by Δ the generalized I × I matrix over A each of whose entries in the diagonal positions is the unity of A and 0 elsewhere. Obviously, Δ is cancellable in M ( A , I , I ) . It is easy to see that the following corollary is an easy consequence of Proposition 2.8.

Corollary 2.9

Let A be a ring with unity. Then A is prime (semiprime, primitive and semiprimitive, respectively) if and only if for any [for some] nonempty set I , M ( A , I , I ; Δ ) is prime (semiprime, primtive and semiprimitive, respectively).

Let us turn back to the proof of Theorem 2.7. Assume now that M is semiprime and the condition:

If q m j , q n k are unit entries of Q , then a q m j ◇ A q n k ◇ b ≠ 0 for any nonzero elements a ∈ q m j ◇ A q m j ◇ , b ∈ q n k ◇ A q n k ◇ .

In this case, u in (2.5) exists in M . Moreover, we can derive Conditions (a) and (b) in the proof of Theorem 2.7, and whence (2.6). So, we have indeed proved the following theorem.

Theorem 2.10

Let M ( A , I , M ; Q ) be a generalized Munn ring. If Q is regular in A , then M is prime if and only if M is semiprime and (PM) is satisfied.

Comparing with Theorem 2.10, it raises a natural conjecture as follows:

Conjecture 2.11

Let M ( A , I , M ; Q ) be a generalized Munn ring, and assume that Q is regular in A . Then the following conditions are equivalent:

M is primitive;
M is semiprimitive and (PM) is satisfied;
M is both semiprimitive and prime.

2.2 Principal ideal domains

In this subsection, we study the primeness of Munn rings over a principal ideal domain. We first provide one property of cancellable matrices over a principal ideal domain.

Lemma 2.12

Let A be a principal ideal domain, and Q a m × n matrix over A . Then Q is cancellable in M ( A , n , m ) if and only if m = n = r Q , where r Q is the rank of Q .

Proof

(Necessity). Assume that Q is cancellable in M ( A , I , M ) . We suppose contrariwise that m = n = r Q is not valid. By [23, Proposition III.2.11], there exist invertible matrices U , V such that

U Q V = D 0 0 0 ,

where D is a diagonal r Q × r Q matrix with nonzero diagonal entries. So, there exists a nonzero matrix A 22 over A such that

0 0 0 A 22 U Q V = 0 .

Moreover,

0 0 0 A 22 U Q = 0 ,

so that

0 0 0 A 22 = 0

since, by hypothesis, Q is cancellable in M ( A , I , M ) . It follows that A 22 = 0 . It is a contradiction. Thus, m = r Q = n .

(Sufficiency). If m = r Q = n , then by [23, Proposition III.2.11], there exist invertible matrices U , V such that U Q V = diag ( d 1 , d 2 , … , d m ) , where d i ≠ 0 for i = 1 , 2 , … , m . For any X = ( x i j ) ∈ M n ( A ) , we have

X Q = 0 ⇔ ( X U − 1 ) U Q V = ( y i j d j ) = 0 , where X U − 1 = ( y i j ) ; ⇔ y i j d j = 0 for i , j = 1 , 2 , … , n ; ⇔ y i j = 0 for i , j = 1 , 2 , … , n ; ⇔ X U − 1 = 0 ; ⇔ X = 0 ,

and similarly, Q X = 0 if and only if X = 0 . Therefore, Q is cancellable in M ( A , m , n ) .□

Proposition 2.13

Let M ( A , m , n ; Q ) be a Munn ring. If A is a principal ideal domain with unity 1, then the following conditions are equivalent:

M is semiprime;
m = r Q = n ;
M is prime.

Proof

(i) ⇒ (ii). By Remark 2.6, Q is cancellable in M ( A , m , n ) . Now Lemma 2.12 results (ii).

(ii) ⇒ (iii). Let A 1 and A 2 be an arbitrary nonzero n × n matrices over A . By [23, Proposition III.2.11], there exist invertible matrices U i , V i , i = 1 , 2 such that

U i A i V i = diag ( d 1 ( i ) , d 2 ( i ) , … , d m i ( i ) , 0 , … , 0 ) ,

where d k ( i ) ≠ 0 for any 1 ≤ k ≤ m i . Let U and V be invertible matrices such that Q = U diag ( c 1 , c 2 , … , c n ) V , where c j ≠ 0 for j = 1 , 2 , … , n . Because

V 1 − 1 U = ( x i j ) , V U 2 − 1 = ( y i j )

are both invertible, there exist 1 ≤ j 0 , i 0 ≤ n such that x 1 j 0 ≠ 0 , y i 0 1 ≠ 0 . Compute

( 1 ) 11 U 1 A 1 Q V − 1 ( 1 ) j 0 i 0 U − 1 Q A 2 V 2 ( 1 ) 11 = ( 1 ) 11 ( U 1 A 1 V 1 ) ( V 1 − 1 U ) diag ( c 1 , … , c n ) ( 1 ) j 0 i 0 ( U − 1 Q V − 1 ) ( V U 2 − 1 ) ( U 2 A 2 V 2 ) ( 1 ) 11 = ( 1 ) 11 diag ( d 1 ( 1 ) , … , d m 1 ( 1 ) , 0 ⋯ , 0 ) ( x i j ) diag ( c 1 , c 2 , … , c n ) ⋅ ( 1 ) j 0 i 0 diag ( c 1 , c 2 , … , c n ) ( y i j ) diag ( d 1 ( 2 ) , … , d m 2 ( 2 ) , 0 ⋯ , 0 ) ( 1 ) 11 = ( d 1 ( 1 ) x 1 j 0 c j 0 c i 0 y i 0 1 d 1 ( 2 ) ) 11 ≠ 0 ,

so that the entry in the ( 1 , 1 ) position of

U 1 A 1 Q V − 1 ( 1 ) j 0 i 0 U − 1 Q A 2 V 2 = U 1 A 1 ∘ V − 1 ( 1 ) j 0 i 0 U − 1 ∘ A 2 V 2

is equal to the nonzero element d 1 ( 1 ) x 1 j 0 c j 0 c i 0 y i 0 1 d 1 ( 2 ) . It follows that A 1 ∘ V − 1 ( 1 ) j 0 i 0 U − 1 ∘ A 2 ≠ 0 since A is a principal ideal domain, giving d 1 ( 1 ) x 1 j 0 c j 0 c i 0 y i 0 1 d 1 ( 2 ) ≠ 0 . Thus, A 1 ∘ M ∘ A 2 ≠ 0 and whence M is prime.

(iii) ⇒ (i). It is obvious.□

Notice that for a principal ideal domain, the unity is the only nonzero idempotent. We observe that a generalized matrix over a principal ideal domain is regular if and only if each of its rows and each of its columns contain at least one unit. By Propositions 2.8 and 2.13, the following corollary is immediate from that any domain is prime.

Corollary 2.14

Let M ( A , m , n ; Q ) be a Munn ring. If

A is a principal ideal domain;
Q is regular in A ,

then M is prime if and only if m = r Q = n .

3 Generalized matrix rings

In this section, we shall consider the primeness and the primitivity of generalized matrix rings. We first recall the definition of generalized matrix rings.

Let I be a nonempty set. For any i , j , l ∈ I , let A i i be a ring with identity 1 i , and A i j a unitary ( A i i , A j j ) -bimodule. Assume that there is a module homomorphism μ i j l : A i j × A j l into A i l , written μ i j l ( x , y ) = x y , for any i , j , l ∈ I . If the following conditions hold:

( x + y ) z = x z + y z , w ( x + y ) = w x + w y ;
w ( x z ) = ( w x ) z ,

for any x , y ∈ A i j , z ∈ A j l , w ∈ A l i , then the triple ( A i j , I , μ i j l ) is called a Γ -system with index I .

Given a Γ -system ( A i j , I , μ i j l ) , let Gℳ = Gℳ ( A i j , I , μ i j l ) be the external direct sum of { A i j : i , j ∈ I } . We shall use { x i j } to denote the external direct sum of x i j with i , j ∈ I . Now we define the multiplication in Gℳ as

x y = ∑ k x i k y k j ,

where x = { x i j } and y = { y i j } . It is easy to check that Gℳ is a ring (possibly without unity). We call Gℳ a generalized matrix ring, or a gm ring for short, written Gℳ ( A i j , I , μ i j k ) or Gℳ for short. If e i i is a nonzero element of A i i satisfying that e i i x = x = x e j j for all x ∈ A i j , then the set { e i i : i ∈ I } is called a generalized matrix unit of the Γ -system ( A i j , I , μ i j u ) (for example, see [14]).

In what follows, we still write the element x = { x i j } satisfying that x i j = 0 if i ≠ i 0 , j ≠ j 0 and x i 0 j 0 = u as { u } i 0 j 0 , especially, write { 1 i 0 } i 0 i 0 = 1 i 0 . Also, we use { A } i j to stand for the set { { a } i j : a ∈ A } . And, we write x = { x i j } i ∈ A , j ∈ B if x i j = 0 whenever i ∈ I \ A or j ∈ I \ B . It is easy to check that the set { 1 i : i ∈ I } is a generalized matrix unit of the Γ -system ( A i j , I , μ i j u ) .

Proposition 3.1

The generalized matrix ring Gℳ ( A i j , I , μ i j k ) is a subring of the generalized Munn ring M ( Gℳ , I , I ; Ξ ) , where Ξ is the generalized I × I matrix in which any entry in the ( i , i ) position is 1 i , for any i ∈ I , and 0 elsewhere.

Proof

Consider the mapping

ϕ : Gℳ → M ( Gℳ , I , I ; Ξ ) ; { x i j } ↦ ∑ ( x i j ) i j .

A routine calculation shows that ϕ is an injective homomorphism, and here, we omit the detail.□

We can now describe the main results of this section.

Theorem 3.2

Let Gℳ = Gℳ ( A i j , I , μ i j l ) be a generalized matrix ring. Then Gℳ is semiprime (semiprimitive, respectively) if and only if the following conditions are satisfied:

for any i , j ∈ I , if x is a nonzero element in A i j , then x A j i ≠ 0 and A j i x ≠ 0 ;
for any i ∈ I , A i i is semiprime (semiprimtive, respectively).

Proof

A routine calculation shows that { 1 i } i i is an idempotent, for all i ∈ I , and the mapping φ : { x } i i ↦ x is an isomorphism from { A i i } i i onto A i i . Compute

{ 1 i } i i Gℳ { 1 i } i i = { A i i } i i .

So,

(3.1) { 1 i } i i Gℳ { 1 i } i i ≅ A i i .

(3.2.1) The proof for the semiprime case. If Gℳ is semiprime, then as for all i ∈ I , { 1 i } i i is an idempotent in Gℳ , we obtain that { 1 i } i i Gℳ { 1 i } i i is semiprime, so that by (3.1), A i i is semiprime.

We contrariwise let i 0 , j 0 ∈ I such that A i 0 j 0 ≠ 0 but A j 0 i 0 = 0 . Pick a nonzero element a in A i 0 j 0 . Of course, { a } i 0 j 0 is a nonzero element in Gℳ . Compute

(3.2) { a } i 0 j 0 Gℳ { a } i 0 j 0 = { a } i 0 j 0 { A j 0 i 0 } j 0 i 0 { a } i 0 j 0 = { a A j 0 i 0 a } i 0 j 0 ⊆ { A j 0 i 0 } i 0 j 0 = 0 ,

contrary to the hypothesis that Gℳ is semiprime. Therefore we have now proved that for any i , j ∈ I ,

(3.3) A i j ≠ 0 ⇔ A j i ≠ 0 .

To see (i), we assume contrarily that x is a nonzero element in A i 1 j 1 such that x A j 1 i 1 = 0 . By (3.2), we have { x } i 1 j 1 Gℳ { x } i 1 j 1 = { x A j 1 i 1 x } i 1 j 1 = 0 , contrary to the hypothesis that Gℳ is semiprime. So, x A j i ≠ 0 . Dually, we may prove that A j i x ≠ 0 .

For the converse, assume that given Conditions (i) and (ii) hold. We oppositely let u = { u i j } be a nonzero element in Gℳ such that u Gℳ u = 0 . Notice that

Ξ is regular in Gℳ and cancellable in M ( Gℳ , I , I ) ;
{ 1 i } i i , i ∈ I are all unit entries of Ξ . Obviously, { 1 i } i i ◇ = { 1 i } i i and furthermore, { 1 i } i i Gℳ { 1 j } j j = { A i j } i j . Together with Condition (i), it is easy to see that Condition (ii) in Theorem 2.5.

By Theorem 2.5, the generalized Munn ring M ( Gℳ , I , I ; Ξ ) is semiprime. Denote

J = { k ∈ I : u k j ≠ 0 for some j ∈ I , or u i k ≠ 0 for some i ∈ I } .

It is not difficult to check that

ε = ∑ i ∈ J ( 1 i ) i i is an idempotent. Moreover, ε M ( Gℳ , I , I ; Ξ ) ε is semiprime.
ε ϕ ( u ) = ϕ ( u ) = ϕ ( u ) ε , where ϕ has the same meanings as in the proof of Proposition 3.1.

Moreover,

ϕ ( u ) ( ε M ( Gℳ , I , I ; Ξ ) ε ) ϕ ( u ) ⊆ ϕ ( u ) ϕ ( Gℳ ) ϕ ( u ) = ϕ ( u Gℳ u ) = 0 ,

contrary to the foregoing proof that ε M ( Gℳ , I , I ; Ξ ) ε is semiprime. Therefore u = 0 and whence Gℳ is semiprime.

(3.2.2) The proof for the semiprimitive case. Similar as (3.2.1), we may prove the necessity.

For the converse, we contrariwise assume that u is a nonzero element in rad ( Gℳ ) . With notations in (3.2.1), we denote X = { { x i j } i ∈ J , j ∈ J } ⊆ Gℳ . It is easy to see that

X is a subalgebra of Gℳ ;
τ = ∑ i ∈ J { 1 i } i i is an idempotent in Gℳ . Moreover, ϕ ( τ ) = ε , τ u = u = u τ and τ Gℳ τ = X ;
ϕ ( X ) = ε M ( Gℳ , I , I ; Ξ ) ε .

Therefore u ∈ rad ( τ Gℳ τ ) = rad ( X ) . Notice that ϕ is an injective homomorphism. We observe that ε M ( Gℳ , I , I ; Ξ ) ε is isomorphic to X . It follows that

ϕ ( u ) ∈ rad ( ε M ( Gℳ , I , I ; Ξ ) ε ) ,

so that ε M ( Gℳ , I , I ; Ξ ) ε is not semiprimitive. Indeed, by the proof of the converse part in (3.2.1), we can obtain that M ( Gℳ , I , I ; Ξ ) is semiprimitive if for any i ∈ I , A i i is semiprimtive. In this case, ε M ( Gℳ , I , I ; Ξ ) ε is semiprimitive, contrary to the foregoing proof that ε M ( Gℳ , I , I ; Ξ ) ε is not semiprimitive. Consequently, Gℳ is semiprimitive.□

Theorem 3.3

Let Gℳ = Gℳ ( A i j , I , μ i j l ) be a generalized matrix ring. Then Gℳ is prime (primitive, respectively) if and only if the following conditions are satisfied:

for any i , j , k , l ∈ I , if x and y are respectively nonzero elements in A i j and in A k l , then x A j k y ≠ 0 ;
for any i ∈ I , A i i is prime (primtive, respectively).

Proof

(3.3.1) The proof for the prime case. Similar as in (3.2.1), we may prove the necessity.

For the sufficiency, we contrarily assume that u = { u i j } and v = { v k l } are nonzero elements in Gℳ such that u Gℳ v = 0 . Obviously, there are i 0 , j 0 , k 0 , and l 0 ∈ I such that u i 0 j 0 and v k 0 l 0 are neither equal to 0. Further, by Condition (i), there is x ∈ A j 0 k 0 such that u i 0 j 0 x v k 0 l 0 ≠ 0 . By the same reason, we have y ∈ A l 0 i 0 such that u i 0 j 0 x v k 0 l 0 y u i 0 j 0 x v k 0 l 0 ≠ 0 . So that 0 ≠ u i 0 j 0 x v k 0 l 0 y ∈ A i 0 i 0 . Compute

{ u i 0 j 0 x v k 0 l 0 y ⋅ A i 0 i 0 ⋅ u i 0 j 0 x v k 0 l 0 y } i 0 i 0 = { u i 0 j 0 } i 0 j 0 { x v k 0 l 0 y } j 0 i 0 Gℳ { u i 0 j 0 x } i 0 k 0 { v k 0 l 0 } k 0 l 0 { y } l 0 i 0 ⊆ { u i 0 j 0 } i 0 j 0<