A regularity criterion for complete matrix semirings (Q1809984)

Let \((S,+,\cdot)\) be an additively commutative semiring with absorbing zero 0. \(S\) is regular if for every \(a\in S\) the equation \(axa=a\) is solvable in \(S\). It is shown, that for \(n\geq 3\) the semiring \(M_n(S)\) of all \(n\times n\)-matrices over \(S\) is regular iff \(S\) is a regular ring. A semiring \((A,+,\cdot)\) with zero 0 is zero sum free if \(x+y=0\) implies \(x=y=0\). Then the matrix semiring \(M_2(S)\) is regular iff \(S\) is the direct sum of a ring \(R\) and a zero sum free semiring \(A\) which satisfies any of the following equivalent conditions: (1) \(M_2(A)\) is regular, (2) \(TXT=T\) is solvable for every upper (lower) triangular matrix \(T\in M_2(A)\), (3) \(A\) is regular and the set \(E(A)\) of idempotents of \((A,\cdot)\) is a distributive lattice with zero and relative complements, (4) \(A\) is regular, all \(e\in E(A)\) are central, and, for every \(e\neq 0\) in \(E(A)\), the ideal \(eA\) is a Boolean idempotent arp-semiring, (5) \(A\) is regular, for every \(e\neq 0\) in \(E(A)\) the semiring \(eAe\) is a Boolean idempotent arp-semiring, and \(\forall e_1,e_2\in E(A),\exists e,f\in E(A)\) such that \(ee_i=e_if=e_i\). -- By a Boolean idempotent arp-semiring \((A,+,\cdot)\) the author means a regular semiring with identity 1 which is additively idempotent, all \(e\in E(A)\) are central, the elements \(1+x\) are invertible and for every \(e\in E(A)\) there is some \(e'\in E(A)\) such that \(e+e'=1\) and \(ee'=0\).