The main aim of the paper is to give a socle theory for Leavitt path algebras of arbitrary graphs. We use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it, respectively. Leavitt path algebras with nonzero socle are described as those which have line points, and it is shown that the line points generate ...

First, we construct a bijection between the set of h-vectors and the set of socle-vectors of artinian algebras. As a corollary, we find the minimum codimension that an artinian algebra with a given socle-vector can have. Then, we study the main problem in the paper: determining when there is a unique socle-vector for a given h-vector. We solve the problem completely if the codimension is at mos...

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$. In this paper, we compute socle of $\cb$-bounded strongly stable ideals and determine that saturation number equigenerated ideals. We also provide explicit formulas for $\sat(I)$ Veronese type $I$. Using formula, show $\sat(I^k)$ is quasi-linear from beginning function explicitly.

SOCLE is a hybrid representation system in which cells of constraints are identified with slots of frame networks. Constraint formulas are maintained with respect to slots in frame networks and in turn provide for the dependency regulation of values on the frames. This paper illustrates the use of SOCLE and outlines the control structure decisions made for its design and implementation.

We show that the concept of strong Noether position for a polynomial ideal I CP is equivalent to δ-regularity and thus related to Pommaret bases. In particular, we provide explicit Pommaret bases for two of the ideal sequences used in Hashemi’s definition of strong Noether position and alternative proofs for a number of his statements. Finally, we show that one consequence of δ-regularity is th...

If X is a Tychonoff space, C(X) its ring of real-valued continuous functions. In this paper, we study non-essential ideals in C(X). Let a be a infinite cardinal, then X is called a-Kasch (resp. ā-Kasch) space if given any ideal (resp. z-ideal) I with gen (I) < a then I is a non-essential ideal. We show that X is an א0-Kasch space if and only if X is an almost P -space and X is an א1-Kasch space...

Let R=k[x,y,z] be a standard graded 3-variable polynomial ring, where k denotes any field. We study grade 3 homogeneous ideals I⊆R defining compressed rings with socle k(−s)ℓ⊕k(−2s+1), s⩾3 and ℓ⩾1 are integers. The case for ℓ=1 was previously studied in [8]; generically minimal resolution constructed all such ideals. paper [7] generalizes this the guise of (iterated) trimming complexes. In pape...