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

2.1 Representation of derivations of matrix rings

Throughout the discussion, unless otherwise mentioned, R denotes an associative (not necessarily commutative) ring. Recall that a derivation of R is an additive map d : R → R such that d ( x y ) = d ( x ) y + x d ( y ) where x , y ∈ R .

For a fixed a ∈ R the map d a ( x ) = [ a , x ] = a x − x a for any x ∈ R is a derivation called inner derivation of R determined by a .

As a consequence of Skolem-Noether theorem, see the study by Herstein ([19], p. 99), an arbitrary derivation of the ring of square matrices over a field is inner.

Let M be a ring of matrices over R . It is a well-known fact that the map δ : M → M such that δ ˜ ( A ) = ( δ ( a i j ) ) for any matrix A = ( a i j ) ∈ M , 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 n ( R ) 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 n ( R ) :

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:

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 N T n ( F ) consisting of matrices over a field F whose entries are zeroes on and over the main diagonal. They proved that any automorphism on N T n ( F ) 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 R n ( K , J ) = N T n ( K ) + M n ( J ) , 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 N T n ( K ) : (1) for each diagonal matrix d = ∑ i = 1 n d i e i i , where d i ∈ K , I d ( A ) = [ d , A ] 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 s of N T n ( K ) is called a strongly nilpotent derivation if for all x ∈ ( N T n ( K ) ) k , it follows that s ( x ) ∈ ( N T n ( K ) ) k + 1 . The main result of the study by Chun and Park [30] is

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 C f ( R ) 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 C f ( R ) generated by the derivation δ of R . The first main result is the following:

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:

Following the theorem, the author drew two conclusions.

A local derivation of an algebra A is defined as a linear map D : A → A such that for each x ∈ A there is a derivation D x : A → A and D ( x ) = D x ( x ) , see for details Kadison [37]. Local automorphisms are defined similarly. 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).

2.2 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 n + c e 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 number field Q [ d ] , 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 . a n n ( 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.

In 2016, Facchini and Leroy [47] generalized the results of Salce and Zanardo’s paper [41] in noncommutative setting.

Recently, the idempotent matrices over noncommutative rings have been discussed by Hou [48], Drensky [49], and Wright [50].

Very recently, in 2024, Vladeva presented a new formula of all semicentral idempotents of upper triangular matrix rings [51].

3.1 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 e x e = x e (resp., e x e = e x ) 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 representation if it is ring isomorphic to a generalized triangular matrix ring

where each diagonal element R i , 1 ≤ i ≤ n , is a ring with a unity, R i j is a left R i -right R j -bimodule for 1 ≤ i < j ≤ n , and the matrices obey the usual rules for matrix addition and multiplication. Let J ℓ ( R ) and J r ( R ) denote the sets of left and right semicentral idempotents of R . For some idempotent, e ∈ R it follows that J ℓ ( e R e ) = { 0 , e } if and only if J r ( e R e ) = { 0 , e } . 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 ordered set { b 1 , … , b n } of nonzero distinct idempotents in R is called a set of left triangulating idempotents of R if all of the following holds:

• b 1 + ⋯ + b n = 1 ;

• b 1 ∈ J ℓ ( R ) ;

• b k + 1 ∈ J ℓ ( c k R c k ) , where c k = 1 − ( b 1 + ⋯ + b k ) for 1 ≤ k ≤ n − 1 .

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:

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:

An idempotent e ∈ R is called inner Peirce trivial if e R ( 1 − e ) R e = 0 and is called outer Peirce trivial if 1 − e 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-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 e R e is an m -Peirce ring for some m , 1 ≤ m < n , and ( 1 − e ) R ( 1 − e ) is an ( n − m ) -Peirce ring. An idempotent e ∈ R is called an n -Peirce idempotent if e R e is an n -Peirce ring.

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:

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.

3.2 Derivations of matrix rings. Links to Amitsur’s theorem

In 2022, Vladeva [59] offers a description of the R -derivations of UTM n ( R ) , the ring of upper triangular matrices over an associative ring R with an identity.

A derivation d of the ring UTM n ( R ) is called R -derivation if d ( λ A ) = λ d ( A ) , where λ ∈ R and A ∈ UTM n ( R ) .

The author considered the matrices ℓ k = e 11 + ⋯ + e k k , where 1 ≤ k ≤ n , and proved that ℓ k are left semicentral idempotents of the ring UTM n ( R ) (the matrix e i j has ( i , j ) -entry 1 and rest zero is called a matrix unit). The inner derivations d ℓ k defined by d ℓ k ( A ) = [ ℓ k , A ] , 1 ≤ k ≤ n − 1 , for any matrix A ∈ UTM n ( R ) are idempotents and linearly independent in the additive group of R -derivations of the ring U T M ( R ) . The author preferred to work with the derivations δ i such that δ i ( A ) = [ e i i , A ] for any A ∈ UTM n ( R ) , i = 1 , … , n . Then d ℓ k ( A ) = ∑ i = 1 k δ i ( A ) . Let D be the additive group of derivations generated by δ 2 , … , δ n . Since δ 2 , … , δ n are R -derivations, it follows that D is an R -module. Since δ 1 + δ 2 + ⋯ + δ n = 0 it follows that δ 1 ∈ D . The derivations δ 2 , … , δ n form a basis of the R -module D . The main result of the paper follows that all derivations of UTM n ( R ) belong to D .

Since d ℓ k = ∑ i = 1 k δ i ∈ D , it appears that the result similar to the aforementioned theorem, when on the right side of the equality d ℓ k 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 .

In 2023, Vladeva [60] investigated the class of endomorphisms α of a ring UTM n ( R ) 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, ℓ + r = 1 , then d ℓ ( x ) = ℓ x r for any x ∈ R define a derivation of R . From the second proposition without the restriction ℓ + r = 1 , it follows that Φ ( x ) = r x 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 ℓ k ∈ UTM n ( R ) [59] the author proves that α k ( A ) = A ℓ k , 1 ≤ k ≤ n , for any matrix A ∈ UTM n ( R ) defined an endomorphism of UTM n ( R ) . Similarly, r m = e n − m + 1 n − m + 1 + ⋯ + e n n , where 1 ≤ m ≤ n is a right semicentral idempotent of UTM n ( R ) and β m ( A ) = r m A , where A ∈ UTM n ( R ) , defines an endomorphism of UTM n ( R ) .

The multiplicative semigroup ℰ n ( R ) generated by all α k and β m , k , m = 1 , … , n , is a commutative semigroup with an identity. The first main result of the article is as follows:

An endomorphism α is called a (0,1)-endomorphism if α ( e i j ) is a (0,1)-matrix. An endomorphism α is called regular if α ( e i i ) = e i i or α ( e i i ) = 0 for all i = 1 , … , n . All endomorphisms that belong to ℰ n ( R ) are regular (0,1)-endomorphisms. The second important result is presented as follows:

4.1 Applications of additively idempotent semirings

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

4.2 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.

We point out, in particular, the book [101] and the recent study [102] by Rowen. All other papers are presented in the previous studies [103–110,112–129].

4.3 Derivations of polynomial semirings. 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, ( x 2 + 1 ) ( x + 1 ) = x 3 + x 2 + x + 1 = ( x + 1 ) 3 are two different factorizations of the polynomial x 3 + x 2 + x + 1 .

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 ℳ 2 ( S ) = a b 0 a a , b ∈ S . Then δ defined by δ a b 0 a = 0 b 0 0 is a derivation. Hence, ℳ 2 ( S ) is a semiring with derivation δ . Note that the semiring S need not be additively idempotent.

Since S may be identified with subsemiring of ℳ 2 ( S ) consisting of matrices of the form a 0 0 a , it follows that every semiring can be embedded in a semiring with nontrivial derivation.

In 2020, Vladeva [134] investigated derivations of the polynomial semiring S [ x ] , where S is a commutative additively idempotent semiring. By using that the map δ 1 : S [ x ] → S [ x ] such that δ 1 ( P ( x ) ) = a m x m − 1 + ⋯ + a 2 x + a 1 for P ( x ) = a m x m + ⋯ + a 2 x 2 + a 1 x + a 0 ∈ S [ x ] is a derivation, the author constructed a new derivation δ f ( x ) : S [ x ] → S [ x ] defined by δ f ( x ) ( P ( x ) ) = δ 1 ( P ( x ) ) f ( x ) for a fixed polynomial f ( x ) ∈ S [ x ] . 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:

For an arbitrary derivation d of S [ x ] and polynomial P ( x ) = a m x m + ⋯ + a 1 x + a 0 , the derivatives d ( a i ) are in general polynomials. Then it follows that the map Δ d : S [ x ] → S [ x ] such that Δ d ( P ( x ) ) = d ( a m ) x m + ⋯ + d ( a 1 ) x + d ( a 0 ) 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 x a = a x + 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 δ her : S [ x ] → S [ x ] such that δ her ( P ( x ) ) = δ ( a 0 ) + δ ( a 1 ) x + ⋯ + δ ( a m ) x m , where P ( x ) = a 0 + a 1 x + ⋯ + a m x m ∈ S [ x ] , which is a derivation of S [ x ] .

Now if d x ( P ( x ) ) = x P ( x ) for P ( x ) ∈ S [ x ] , it follows that d x ( P ( x ) ) = δ her ( P ( x ) ) + P ( x ) x . This equality implies that d x is a derivation of S [ x ] .

If C ( S ) is the center of S and C o δ ( S ) is the subsemiring of S , consisting the constants δ , then Γ ( S ) = C ( S ) ∩ C o δ ( S ) is a nontrivial subsemiring of S . The author proved that d x k ( P ( x ) ) = x k P ( x ) and for α ∈ Γ ( S ) the map d α x k ( P ( x ) ) = α x k P ( x ) are derivations of S [ x ] and then state the result as follows:

A derivation d : S [ x ] → S [ x ] such that d ( δ ( a ) ) = δ her ( d ( a ) ) , where a ∈ S , is called a δ - derivation. For each δ -derivation d : S [ x ] → S [ x ] , the map Δ d defined by Δ d ( P ( x ) ) = d ( a 0 ) + d ( a 1 ) x + ⋯ + d ( a m ) x m , where P ( x ) = a 0 + a 1 x + ⋯ + a m x m , is a derivation of S [ x ] .

Vladeva [134] showed that ∂ defined by ∂ ( P ( x ) ) = a 1 + a 2 x + ⋯ + a m x m − 1 is a derivation of S [ x ] .

The main result is similar to Amitsur’s theorem:

4.4 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 2017, Vladeva [139] studied derivatives of triangular, Toeplitz, and circulant matrices over an additively idempotent semiring.

In 2020, Tan [140] obtained a condition for an idempotent matrix over a commutative semiring to be diagonalizable.

In 2022, Vladeva [141] explored the semiring UTM n ( S ) of upper triangular matrices over an additively idempotent semiring S . By using the sums D ¯ k = e 11 + ⋯ + e k k , the author constructed a derivation δ k such that δ k ( A ) = D ¯ k A , where A ∈ UTM n ( S ) . Similarly by D ̲ m = e n − m + 1 n − m + 1 + ⋯ + e n n is built a derivation d m ( A ) = A D ̲ m , where A ∈ UTM n ( S ) . But D ¯ k and D ̲ m are just the left and right semicentral idempotents, respectively, considered in [59] (see Subsection 3.2). The set D of derivations δ k and d m is an additively idempotent semiring such that for the products δ k d m that the author proved

Moreover, D is an S -semimodule and the derivations δ k d n − k , k = 1 , … , n − 1 , and δ k d n − k + 1 , k = 1 , … , n , are the elements of the basis of D .

In the central conclusion of the paper, we obtain the Amitsur’s idea:

In 2021, Vladeva [142] studied the semiring M n ( S ) 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.

As a consequence in the commutative case is obtained that any S -derivation of the matrix semiring M n ( S ) is a hereditary derivation:

The element g ∈ S is called a generator of the derivation D .

The set of left (resp., right) semicentral idempotent elements of S is denoted by L O ( S ) (resp., R O ( S ) ). For x ∈ L O ( S ) (resp., x ∈ R O ( S ) ), the map d x ℓ : S → S (resp., d x r : S → S ) such that d x ℓ ( a ) = x a (resp., d x r ( a ) = a x ) for a ∈ S is a derivation in S .

The derivation D : M n ( S ) → M n ( S ) is called a left (resp., right) derivation if g ∈ L O ( S ) (resp., g ∈ R O ( 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]: