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Let $R$ be a commutative additively idempotent semiring. In this paper, some properties and characterizations for permanents of matrices over $R$ are established, and several inequalities for permanents are given. Also, the adjiont matrices of matriecs over $R$ are considered. Partial results obtained in this paper generalize the corresponding ones on fuzzy matrices, on lattice matrices and on incline matrices.
In this paper we deal with the problem of computing the sum of the $k$-th powers of all the elements of the matrix ring $mathbb{M}_d(R)$ with $d>1$ and $R$ a finite commutative ring. We completely solve the problem in the case $R=mathbb{Z}/nmathbb{Z}
In the well-known construction of the field of fractions of an integral domain, division by zero is excluded. We introduce fracpairs as pairs subject to laws consistent with the use of the pair as a fraction, but do not exclude denominators to be zer
A $d$-dimensional matrix is called emph{$1$-polystochastic} if it is non-negative and the sum over each line equals~$1$. Such a matrix that has a single $1$ in each line and zeros elsewhere is called a emph{$1$-permutation} matrix. A emph{diagonal} o
Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if and only if each $R$-algebra which is a direct limit of module finite $R$-algebras is strongly clean. So, the matrix ring $mathbb{M}_n(R)$ is strongly clean for each integer $
We characterize derivations and 2-local derivations from $M_{n}(mathcal{A})$ into $M_{n}(mathcal{M})$, $n ge 2$, where $mathcal{A}$ is a unital algebra over $mathbb{C}$ and $mathcal{M}$ is a unital $mathcal{A}$-bimodule. We show that every derivation