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A system of four matrix equations over von Neumann regular rings and its applications. (English) Zbl 1083.15021
The author considers the following matrix systems over a von Neumann regular ring $$R$$ with unity (i.e. for each $$a\in R$$ there exists $$b\in R$$ such that $$aba=a$$): (*) $$A_1X=C_1$$, $$XB_2=C_2$$, $$A_3XB_3=C_3$$, $$A_4XB_4=C_4$$; (**) $$A_1X=C_1$$, $$A_3X=C_3$$; (***) $$A_1X=C_1$$, $$A_3XB_3=C_3$$. He gives necessary and sufficient conditions for solvability, and he explicits the general solution to system (*), the bisymmetric one to system (**) and the perselfconjugate one to system $$(***)$$, the latter two in the case when there is an involution $$\sigma$$ over $$R$$ (char$$R\neq 2$$). An $$n\times n$$-matrix $$A=(a_{ij})$$ is bisymmetric if $$a_{ij}=a_{n-i+1,n-j+1}=\sigma (a_{ji})$$; it is perselfconjugate if $$A=A^{(*)}$$ where $$A^{(*)}=VA^*V$$, $$V$$ being the permutation matrix with units along the antidiagonal and zeros elsewhere, and $$A^*=(\sigma (a_{ji}))$$.

##### MSC:
 15A24 Matrix equations and identities 15B33 Matrices over special rings (quaternions, finite fields, etc.) 15B57 Hermitian, skew-Hermitian, and related matrices 15A09 Theory of matrix inversion and generalized inverses 16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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