In this work we introduce a new concept, namely, τ s τs -extending modules (rings) which is torsion-theoretic analogues of extending and then extend many results from to concept. For instance show that for any ring R with unit, if $_{R}R$ purely only every cyclic -nonsingular -module flat. Also, make classification the direct sums rings be -extending.

Moduli spaces arise in classification problems in algebraic geometry (and other areas of geometry) when, as is typically the case, there are not enough discrete invariants to classify objects up to isomorphism. In the case of nonsingular complex projective curves (or compact Riemann surfaces) the genus g is a discrete invariant which classifies the curve regarded as a topological surface, but d...

Let X be a complete nonsingular algebraic curve over C and L a line bundle of degree 1 over X. It is well known that the isomorphism classes of stable bundles of rank 2 and determinant L over X form a nonsingular projective variety S(X). The Betti numbers of S(X) are also known. In this paper we define certain distinguished cohomology classes of S(X) and prove that these classes generate the ra...

The abeliant is a polynomial rule which to each n by n by n+2 array with entries in a commutative ring with unit associates an n by n matrix with entries in the same ring. The theory of abeliants, first introduced in an earlier paper of the author, is simplified and extended here. Now let J be the Jacobian of a nonsingular projective algebraic curve defined over an algebraically closed field. W...

We design algorithms for nding roots of polynomials over function elds of curves. Such algorithms are useful for list decoding of Reed-Solomon and algebraicgeometric codes. In the rst half of the paper we will focus on bivariate polynomials, i.e., polynomials over the coordinate ring of the a ne line. In the second half we will design algorithms for computing roots of polynomials over the funct...

We design algorithms for nding roots of polynomials over function elds of curves. Such algorithms are useful for list decoding of Reed-Solomon and algebraic-geometric codes. In the rst half of the paper we will focus on bivariate polynomials, i.e., polynomials over the coordinate ring of the aane line. In the second half we will design algorithms for computing roots of polynomials over the func...

in this paper, we investigate the existence of a positive solution of fully fuzzy linear equation systems. this paper mainly to discuss a new decomposition of a nonsingular fuzzy matrix, the symmetric times triangular (st) decomposition. by this decomposition, every nonsingular fuzzy matrix can be represented as a product of a fuzzy symmetric matrix s and a fuzzy triangular matrix t.

The element $q$ of a ring $R$ is called quasi-idempotent if $q^2=uq$ for some central unit $u$ $R$, or equivalently $q=ue$, where and $e$ an idempotent $R$. In this paper, we define that the almost quasi-clean each sum regular element. Several properties almost-quasi clean rings are investigated. We prove every quasi-continuous nonsingular quasi-clean. Finally, it determined conditions under wh...

The Kerr black holes possess a photon region with prograde and retrograde orbits radii, respectively, $M\leq r_p^-\leq 3M$ $3M\leq r_p^+\leq 4M$, thereby always cast closed ring or shadow silhouette for $a\leq M$. For $a>M$, it is no-horizon spacetime (naked singularity) wherein spiral into the central singularity, produce an arc-like dark spot at center. We compare holes' structure those produ...