In this paper we investigate the conditions under which a given ring is a sub-ring of a semi-simple ring. For convenience, we say that a ring A is an extension of a ring B, if B is a sub-ring of A. It is found that the existence of a semi-simple extension is equivalent to the vanishing of the extension radical, a two-sided ideal defined analogously to the ordinary radical. In Theorem II we give...

For an unmixed bipartite graph G we consider the lattice of vertex covers LG and compute depth, projective dimension and extremal Bettinumbers of R/I(G) in terms of this lattice.

A new algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. By the use of the Chinese Remainder Theorem (CRT), or of the Discrete Fourier Transform (DFT), that ring can be decomposed into a direct product of fields. That ring decomposition in turn yields a code construction from codes of lower ...

We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.

A ΣΠΣΠ(k) circuit C = ∑k i=1 Fi = ∑k i=1 ∏di j=1 fij is unmixed if for each i ∈ [k], Fi = fi1(x1) · · · fin(xn), where each fij is a univariate polynomial given in the sparse representation. In this paper, we give a polynomial time black-box algorithm of identity testing for the low degree unmixed ΣΠΣΠ(k) circuits. In order to obtain the black-box algorithm, we first show that a special class o...

In the present paper new criteria for classes of FP -injective and weakly quasi-Frobenius rings are given. Properties of both classes of rings are closely linked with embedding of finitely presented modules in fp-flat and free modules respectively. Using these properties, we describe classes of coherent CF and FGF-rings. Moreover, it is proved that the group ring R(G) is FP -injective (resp. we...