This paper aims to investigate and categorize all inequivalent irreducible Z+-modules of a commutative unit Z+-ring A, equipped with set {1, x, y, xy} satisfying x2=1,y2=1 as Z+-basis by using matrix equations, which was part call for Special Issue about inequalities equations Symmetry. If the rank Z+-module n≤2, we prove that there are finitely many Z+-modules, respectively, one three. However...

Let $\Delta =\left(\begin{smallmatrix} A & {_AN_B}\\ {_BM_A} B \\\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_{A}N=0=N\otimes_{B}M$.We first study how to construct (complete) duality pairs of $\Delta$-modules using $A$-modules and $B$-modules, generalizing the result Mao (Comm. Algebra, 2020, 12: 5296--5310) about over triangular matrix ring. Moreover, we Gorenstein projective modu...

In this paper, we give necessary and sufficient conditions for the matrix [ a 0 b d ] , over a *-regular ring, to have a Moore-Penrose inverse of four different types, corresponding to the four cases where the zero element can stand. In particular, we study the case where the MoorePenrose inverse of the matrix flips. Mathematics subject classification (2010): 15A09, 16E50, 16W10.

The paper studies the characterization of adjacency matrix corresponding to zero-divisor graphs of finite commutative ring of Gaussian integer under modulo „n‟. For each positive integer we calculate number of zerodivisors & examine nature of the matrix, and then we generalized the order of matrix in each case. Firstly, we have started with some example, which motivates the later results. The s...

In the ring of symmetric functions the inverse Kostka matrix appears as the transition matrix from the bases given by monomial symmetric functions to the Schur bases. We present both a combinatorial characterization and a recurrent formula for the entries of the inverse Kostka matrix which are different from the results obtained by Egecioglu and Remmel [ER] in 1990. An application to the topolo...

We give a polynomial bound on the spectral density function of a matrix over the complex group ring of Zd. It yields an explicit lower bound on the Novikov-Shubin invariant associated to this matrix showing in particular that the Novikov-Shubin invariant is larger than zero.

A general conjecture is given for an explicit basis of the coordinate ring of the closure of the conjugacy class of a nilpotent matrix. This conjecture is proven when the partition given by the transpose Jordan type of the nilpotent matrix is a hook or has two parts.

The slippery potential energy surface of aryl nitrenes has revealed unexpected and fascinating reactions. To explore such a challenging surface, one powerful approach is to use a combination of a cryogenic matrix environment and a tunable narrowband radiation source. In this way, we discovered the heavy-atom tunneling reaction involving spontaneous ring expansion of a fused-ring benzazirine int...

Within a few years it will be possible to integrate a billion transistors on a single chip. At this integration level, we propose using a high level ISA to express parallelism to hardware instead of using a huge transistor budget to dynamically extract it. Since the fundamental data structures for a wide variety of applications are scalar, vector, and matrix, our proposed Trident processor exte...