Invertible Matrices in Certain Commutative Subsemirings of Full Matrix Semirings

Home , Noncommutative ring, Semifield, Subring

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 commutative 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.

This shows that AB(= BA) = In. Hence A is invertible.

Example 2.4. Let n > 1 and

n+1 1 0 ··· 0 2 + 1 n+1 0 2 2 + 2 0  . .  ....    n−1 0 n   2   . .  A= . 0 n+1 0 .   2   n 0 n−1   2   ..   .. ..     0 n+1 + 2 2 0   2   n+1 + 1 0 ··· 0 1   2    if n is odd, and 196 N. Sirasuntorn, R.I. Sararnrakskul

n 1 0 ··· 0 2 + 1 0 2 n + 2 0  2  ......    . .   .. 0 0 ..     . .   . 0 n n 0 .  B= 2  if n is even.  . n .   . 0 n 0 .   2   .. ..   . 0 0 .     ....   . .   n   0 2 + 2 2 0   n   + 1 0 ··· 0 1   2    Clearly that det A 6= 0 and det B 6= 0, so A and B are invertible over a ﬁeld R [Q]. However, by Theorem 2.3, A and B are not invertible over a semiﬁeld + + (R0 , +, ·) [(Q0 , +, ·)]. If

1 0 0 ······ 0 0 0 0 0 0 ······ 0 2 0   0 0 0 3 0 0  . .   ....    . . . . . . . 0 .. . . C=  , . . . . . . . .. 0 . .    ....   . .    0 0 3 0 0 0   0 2 0 ······ 0 0 0   0 0 0 ······ 0 0 1     + + then C is invertible over the semiﬁeld (R0 , +, ·) [(Q0 , +, ·)], so C is invertible over the semiﬁeld (R, +, ·) [(Q, +, ·)]. In 2014, N. Sirasuntorn and R. I. Sararnrakskul [4] show that the semiring DVn(S) of all A ∈ Mn(S) of the form

x1 0 ··· 0 x1 0 x2 ··· x2 0   ···············  0 x ··· x 0   2 2   x 0 ··· 0 x   1 1    INVERTIBLE MATRICES IN CERTAIN COMMUTATIVE... 197

is a regular commutative subsemiring of the semiring Mn(S) is where S is a regular semiring having some properties. Note that if S is a semiﬁeld then DVn(S) ( Dn(S).

Theorem 2.5. Let S be a semiﬁeld. Then the set DVn(S) is a commutative subsemiring of Dn(S).

Proof. This proof is straightforward.

Corollary 2.6. Let S be a semiﬁeld which is not a ﬁeld. Then every element in DVn(S) is not invertible.

Proof. By Theorem 2.3 and the deﬁnition of the set DVn(S).

Acknowledgement The authors acknowledge the support of the SWU En- dowment Fund Year 2014 through the reseach project, under contract number 112/2557, Srinakharinwirot University.

References [1] R.I. Sararnrakskul, P. Lertwichitsilp, A. Wasanawichit, S. Pianskool, Certain maximal commutative subrings of full matrix rings, Chamchuri J. Math., 2, No. 1 (2010), 47-55. [2] R.I. Sararnrakskul, S. Sombatboriboon, Invertible Matrices over semiﬁelds, East-West J. Math., 11, No. 2 (2009), 223-229. [3] P. Sinutoke, Master Thesis: On the Theory of Semiﬁelds, Chulalongkorn University, Thailand (1980). [4] N. Sirasuntorn, R.I. Sararnrakskul, Regularity of Certain Subsemirings of Full Matrix Semirings, East-West J. Math., 17, No. 1 (2015), 23-32. 198