INVERTIBLE MATRICES IN CERTAIN COMMUTATIVE SUBSEMIRINGS OF FULL MATRIX SEMIRINGS

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 detA 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).

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 = 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 = 0).

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 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 . Note that such a y is unique and may be written as 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, We note that both (R + 0 , +, •) and (Q + 0 , +, •) 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 M n (S) be the set of all n × n matrices over S. Then under the usual addition and multiplication of matrices, M n (S) is also an additively commutative semiring with zero and the n × n zero matrix over S is the zero of the matrix semiring M n (S).For A ∈ M n (S) and i, j ∈ {1, 2, . . ., n}, let A ij be the entry of A in the i th row and the j th column.
In 2010, Sararnrakskul, Lertvijitsilp, Wassanawichit and Pianskool [1] prove is a maximal commutative subring of the ring M n (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 = I n where I n 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 D n (S) is invertible over S where S is a semifield.Moreover, D n (S) is a maximal commutative subsemiring of the semiring M n (S).
By the proof of Lemma 2.2 and Theorem 2.3 in [1], we have more generalized result for D n (S) where S is a semifield as the following theorem.
Theorem 2.2.The set D n (S) is a maximal commutative subsemiring of the semiring M n (S).
It is well-known that a square matrix A over a field F is invertible if and only if det A = 0. Therefore if S is a field and A ∈ D n (S) then A is invertible if and only if det A = 0.For this reason if S is a semifield which is not a field and A ∈ D n (S) when A is invertible.So, we characterize invertible matrices in a commutative subsemiring of the semiring M n (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 ∈ D n (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, A ii = 0 if and only if A i,n−i+1 = 0. and Also if i, j ∈ {1, 2, . . ., n} are such that j = i and j = n − i + 1, then by Note 2.1, we have This shows that AB(= BA) = I n .Hence A is invertible.
Example 2.4.Let n > 1 and if n is odd, and if n is even.
Clearly that det A = 0 and det B = 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 In 2014, N. Sirasuntorn and R. I. Sararnrakskul [4] show that the semiring DV n (S) of all A ∈ M n (S) of the form is a regular commutative subsemiring of the semiring M n (S) is where S is a regular semiring having some properties.Note that if S is a semifield then DV n (S) D n (S).
Theorem 2.5.Let S be a semifield.Then the set DV n (S) is a commutative subsemiring of D n (S).
Proof.This proof is straightforward.
Corollary 2.6.Let S be a semifield which is not a field.Then every element in DV n (S) is not invertible.
Proof.By Theorem 2.3 and the definition of the set DV n (S).