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Invertible Matrices in Certain Commutative Subsemirings of Full Matrix Semirings

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International Journal of Pure and Applied Mathematics Volume 106 No. 1 2016, 191-197 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v106i1.14 ijpam.eu INVERTIBLE MATRICES IN CERTAIN COMMUTATIVE SUBSEMIRINGS OF FULL MATRIX SEMIRINGS N. Sirasuntorn1, R.I. Sararnrakskul2 § 1,2Department of Mathematics Faculty of Science Srinakharinwirot University 114 Sukhumvit 23, Wattana District, Bangkok 10110, THAILAND Abstract: Then every field is a semifield. For a semifield S, we let Dn(S) denote the set of all A ∈ Mn(S) of the form x1 0 ··· 0 y1 3 0 x2 ··· y2 0 ··············· 0 y2 ··· x2 0 y1 0 ··· 0 x1 where Mn(S) is the full n×n matrix semiring over S. Then Dn(S) is a maximal commutative subsemiring of the semiring Mn(S). If S is a field, it is known that A ∈ Dn(S) is invertible if and only if det A =6 0. In this paper, invertible matrices in Dn(S) where S is a semifield which is not a field are characterized. It is shown that if S is a semifield which is not a field, then A ∈ Dn(S) is an invertible matrix over S if and only if (xi = 0 if and only if yi =6 0). AMS Subject Classification: 16Y60, 20M17, 15A09 Key Words: semiring, semifield, full matrix semiring, invertible matrix 1. Introduction and Preliminaries A semiring S is an algebraic structure (S, +, ·) such that (S, +) and (S, ·) are semigroups and · is distributive over +. A semiring (S, +, ·) is called additively Received: Octobere 21, 2015 c 2016 Academic Publications, Ltd. Published: February 3, 2016 url: www.acadpubl.eu §Correspondence author 192 N. Sirasuntorn, R.I. Sararnrakskul [multiplicatively] commutative if x + y = y + x [x · y = y · x] for all x, y ∈ S. We say that (S, +, ·) is commutative if it is both addtitively commutative and multiplicatively commutative. An element 0 of S is called a zero of the semiring (S, +, ·) if x + 0 = x = 0+ x and x ·0 = 0 = 0·x for all x ∈ S and by an identity of (S+, ·) we mean an element 1 ∈ S such that x · 1 = 1 · x = x for all x ∈ S. Note that a zero and an identity of a semiring are unique. If a semiring (S, +, ·) has a zero 0 [an identity 1], we say that an element x ∈ S is additively [multiplicatively] invertible over S if there exists an element y ∈ S such that x + y = y + x = 0 [x · y = y · x = 1]. Note that such a y is unique and may be written as −x [x−1]. A commutative semiring (S, +, ·) with zero 0 and identity 1 is called a semifield if (S \{0}, ·) is a group. Then every field is a semifield. It is clearly seen that the following fact holds in any semifield. Proposition 1.1. If S is a semifield, then for all x, y ∈ S, xy = 0 implies that x = 0 or y = 0. Example 1.2 ([2]). Let R be the set of all real numbers, Q the set of + + + + rational numbers, R = {x ∈ R | x > 0}, R0 = R ∪{0}, Q = {x ∈ Q | x > 0} + + + + and Q0 = Q ∪ {0}. Then (R0 , +, ·) and (Q0 , +, ·) are semifields which are not fields. + + We note that both (R0 , +, ·) and (Q0 , +, ·) are semifields which are not fields. These semifields have the property that 0 is the only additively invertible element, that is, for x, y ∈ S, x + y = 0 implies x = y = 0. In fact, this property is generally true. Proposition 1.3 ([3]). If S is a semifield which is not a field, then 0 is the only additively invertible element of S. A maximal commutative subsemiring of a semiring S is defined naturally to be a maximal element of the set of all proper commutative subsemirings of S under inclusion. If S is a noncommutative ring, then a maximal commutative subsemiring of S is a maximal element of the set of all commutative subsemiring of S under inclusion. For a positive integer n and an additively commutative semiring S with zero, let Mn(S) be the set of all n × n matrices over S. Then under the usual addition and multiplication of matrices, Mn(S) is also an additively commu- tative semiring with zero and the n × n zero matrix over S is the zero of the matrix semiring Mn(S). For A ∈ Mn(S) and i, j ∈ {1, 2, . , n}, let Aij be the entry of A in the ith row and the jth column. In 2010, Sararnrakskul, Lertvijitsilp, Wassanawichit and Pianskool [1] prove INVERTIBLE MATRICES IN CERTAIN COMMUTATIVE... 193 that the ring Dn(R) of all A ∈ Mn(R) of the form x1 0 ··· 0 y1 0 x2 ··· y2 0 ··············· 0 y ··· x 0 2 2 y 0 ··· 0 x 1 1 is a maximal commutative subring of the ring Mn(R) where R is a commutative ring. Let S = (S, +, ·) be a commutative semiring with zero 0 and identity 1. An n × n matrix A over S is called invertible over S if there is an n × n matrix B over S such that AB = BA = In where In is the identity n × n matrix over S. Note that such a B is unique. The purpose of this paper is to show that when a square matrix in Dn(S) is invertible over S where S is a semifield. Moreover, Dn(S) is a maximal commutative subsemiring of the semiring Mn(S). 2. The Subsemiring Dn(S) of Mn(S) n From now on, let S be a semifield. Let n ∈ N \{1} and Λ = {1, 2,..., 2 }. Note 2.1. For A ∈ Mn(S),A ∈ Dn(S) if and only if (i) Aii = An−i+1,n−i+1 and Ai,n−i+1 = An−i+1,i for all i ∈ Λ and (ii) Aij = 0 for all i, j ∈ {1, 2, . , n} with j 6= i and j 6= n − i + 1. By the proof of Lemma 2.2 and Theorem 2.3 in [1], we have more generalized result for Dn(S) where S is a semifield as the following theorem. Theorem 2.2. The set Dn(S) is a maximal commutative subsemiring of the semiring Mn(S). It is well-known that a square matrix A over a field F is invertible if and only if det A 6= 0. Therefore if S is a field and A ∈ Dn(S) then A is invertible if and only if det A 6= 0. For this reason if S is a semifield which is not a field and A ∈ Dn(S) when A is invertible. So, we characterize invertible matrices in a commutative subsemiring of the semiring Mn(S) where S is a semifield which is not a field. Theorem 2.3. Let S be a semifield which is not a field. Then A ∈ Dn(S) is invertible if and only if every row and every column of A contains exactly one nonzero element, that is, for each i ∈ Λ,Aii = 0 if and only if Ai,n−i+1 6= 0. 194 N. Sirasuntorn, R.I. Sararnrakskul Proof. It is evident if n = 1. Assume that n > 1 and A ∈ Dn(S) is invertible. Let B ∈ Dn(S) such that AB = BA = In. Then we have the following equalities for i ∈ Λ, n 1 = (AB)ii = AikBki Xk=1 = AiiBii + Ai,n−i+1Bn−i+1,i (1) and n 0 = (AB)i,n−i+1 = AikBk,n−i+1 Xk=1 = AiiBi,n−i+1 + Ai,n−i+1Bn−i+1,n−i+1 (2) By (2) and Proposition 1.3 we obtain that AiiBi,n−i+1 = 0 and Ai,n−i+1Bn−i+1,n−i+1 = 0 (3) If Aii = 0, then by (1) Ai,n−i+1 6= 0. Assume that Ai,n−i+1 6= 0 and Aii 6= 0. By (3), Bi,n−i+1 = 0 and Bn−i+1,n−i+1 = 0. Since Bii = Bn−i+1,n−i+1 = 0 and Bn−i+1,i = Bi,n−i+1 = 0, by (1) we have AiiBii + Ai,n−i+1Bn−i+1,i = 0 a contradiction. So, we have if Ai,n−i+1 6= 0 then Aii = 0. Conversely, assume that for each i ∈ Λ,Aii = 0 if and only if Ai,n−i+1 6= 0. Define B ∈ Dn(S) by −1 Aij if Aij 6= 0, Bij = (0 if Aij = 0. We will show that AB = In. The following equalities for i ∈ Λ: n (AB)ii = AikBki Xk=1 = AiiBii + Ai,n−i+1Bn−i+1,i = AiiBii + Ai,n−i+1Bi,n−i+1 Ai,n−i Bi,n−i if Aii = 0 = +1 +1 (AiiBii if Aii 6= 0 = 1. INVERTIBLE MATRICES IN CERTAIN COMMUTATIVE... 195 and n (AB)i,n−i+1 = AikBk,n−i+1 Xk=1 = AiiBi,n−i+1 + Ai,n−i+1Bn−i+1,n−i+1 Ai,n−i Bn−i ,n−i if Aii = 0 = +1 +1 +1 (AiiBi,n−i+1 if Aii 6= 0 0 if Aii = 0 (∵ Bn−i ,n−i = Bii = Aii = 0) = +1 +1 (0 if Aii 6= 0 (∵ Bi,n−i+1 = Ai,n−i+1 = 0). Also if i, j ∈ {1, 2, . , n} are such that j 6= i and j 6= n − i + 1, then by Note 2.1, we have n (AB)ij = AikBkj Xk=1 = AiiBij + Ai,n−i+1Bn−i+1,j = Aii0 + Ai,n−i+10 = 0.
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    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.