We use hypersurface support to classify thick (two-sided) ideals in the stable categories of representations for several families finite-dimensional integrable Hopf algebras: bosonized quantum complete intersections, Borels type $A$, Drinfeld doubles height 1 finite characteristic, and rings functions on group schemes over a perfect field. then identify prime ideal (Balmer) spectra these catego...

Let $R$ be a commutative ring and $I$ an ideal of $R$. The zero-divisor graph of $R$ with respect to $I$, denoted by $Gamma_I(R)$, is the simple graph whose vertex set is ${x in Rsetminus I mid xy in I$, for some $y in Rsetminus I}$, with two distinct vertices $x$ and $y$ are adjacent if and only if $xy in I$. In this paper, we state a relation between zero-divisor graph of $R$ with respec...

We present a (random) polynomial-time algorithm to generate a randomGaussian integer with the uniform distribution among those with norm at most N , along with its prime factorization. The method generalizes to finding a random ideal in the ring of integers of a quadratic number field together with its prime ideal factorization. We also discuss the analogous problem for higher degree number fie...

It is shown that a squarefree principal Borel ideal satisfies the persistence property for the associated prime ideals. For the graded maximal ideal we compute the index of stability with respect to squarefree principal Borel ideals and determine their stable set of associated prime ideals.

Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this paper, we will introduce the notions of 2-absorbing $I$-prime and 2-absorbing $I$-second submodules of an $R$-module $M$ as a generalization of 2-absorbing and strongly 2-absorbing second submodules of $M$ and explore some basic properties of these classes of modules.

For certain weak versions of the Axiom of Choice (most notably, the Boolean Prime Ideal theorem), we obtain equivalent formulations in terms of partial orders, and filter-like objects within them intersecting certain dense sets or antichains. This allows us to prove some consequences of the Boolean Prime Ideal theorem using arguments in the style of those that use Zorn’s Lemma, or Martin’s Axiom.

We introduce compactness theorems for generalized colorings and derive several particular compactness theorems from them. It is proved that the theorems and many of their consequences are equivalent in ZF set theory to BPI, the Prime Ideal Theorem for Boolean algebras. keywords: generalized graph colorings, compactness, prime ideal theorem MSC: 05C15; 03E25 .