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 with identity and $N(R)$ and
 $J\left(R\right)$ denote the nilradical Jacobson radical
 of $R$, respectively. A proper ideal $I$ is called an
 n-ideal if for every $a,b\in R$, whenever $ab\in I$\ $a\notin
 N(R)$, then $b\in I$. In this paper, we introduce study
 J-ideals as new generalization n-ideals in rings.
 $I$\ $R$\ J-ideal $ab\i...

Let S(f) denote the set of integral ideals / such that / is a permutation polynomial modulo i", where / is a polynomial over the ring of integers of an algebraic number field. We obtain a classification for the sets S which may be written in the form S(f). Introduction. A polynomial f(x) with coefficients in a commutative ring R is said to be a permutation polynomial modulo an ideal I oi R (abb...

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...

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.

We call a module  essentially retractable if HomR for all essential submodules N of M. For a right FBN ring R, it is shown that: (i)  A non-zero module  is retractable (in the sense that HomR for all non-zero ) if and only if certain factor modules of M are essentially retractable nonsingular modules over R modulo their annihilators. (ii)  A non-zero module  is essentially retractable if and on...

Certainly 0R ∈ A. Let a, b ∈ A. Then a ∈ An and b ∈ Am for some n,m ∈ N. Without loss of generality, assume n ≤ m. This means An ⊆ Am. So we have a, b ∈ Am. Since Am is a subring, it follows that −a ∈ Am, a + b ∈ Am, and ab ∈ Am. So also −a ∈ A, a + b ∈ A and ab ∈ A. This means A is closed under additive inverses, addition, and multiplication, so A is a subring of R. Let r ∈ R. Since An is an i...

We use pseudodeformation theory to study Mazur’s Eisenstein ideal. Given prime numbers N and p > 3, we study the Eisenstein part of the p-adic Hecke algebra for Γ0(N). We compute the rank of this Hecke algebra in terms of Massey products in Galois cohomology, answering a question of Mazur and generalizing a result of Calegari-Emerton. We also also give new proofs of Merel’s result on this rank ...

For every prime number p > 5 it is shown that, under certain hypotheses on x e Q , the imaginary quadratic fields Q( \/x2p 6xf + 1 ) have ideal class groups with noncyclic p-parts. Several numerical examples with p = 5 and 7 are presented. These include the field Q(v/-4805446123032518648268510536). The 7-part of its class group is isomorphic to C(7) x C(7) x C(7), where C(n) denotes a cyclic gr...