Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDynamical ideals of non-commutative rings
70
0
0.0
(
0
)
Authors
Igor Nikolaev
Ask ChatGPT about the research
No Arabic abstract
An analog of the prime ideals for simple non-commutative rings is introduced. We prove the fundamental theorem of arithmetic for such rings. The result is used to classify the surface knots and links in the smooth 4-dimensional manifolds.
rate research
Read More
Let $mathcal{I}(R)$ be the set of all ideals of a ring $R$, $delta$ be an expansion function of $mathcal{I}(R)$. In this paper, the $delta$-$J$-ideal of a commutative ring is defined, that is, if $a, bin R$ and $abin Iin mathcal{I}(R)$, then $ain J(R)$ (the Jacobson radical of $R$) or $bin delta(I)$. Moreover, some properties of $delta$-$J$-ideals are discussed,such as localizations, homomorphic images, idealization and so on.
We study the q-commutative power series ring R:=k_q[[x_1,...,x_n]], defined by the relations x_ix_j = q_{ij}x_j x_i, for multiplicatively antisymmetric scalars q_{ij} in a field k. Our results provide a detailed account of prime ideal structure for a class of noncommutative, complete, local, noetherian domains having arbitrarily high (but finite) Krull, global, and classical Krull dimension. In particular, we prove that the prime spectrum of R is normally separated and is finitely stratified by commutative noetherian spectra. Combining this normal separation with results of Chan, Wu, Yekutieli, and Zhang, we are able to conclude that R is catenary. Following the approach of Brown and Goodearl, we also show that links between prime ideals are provided by canonical automorphisms. Moreover, for sufficiently generic q_{ij}, we find that R has only finitely many prime ideals and is a UFD (in the sense of Chatters).
Let $A$ be any associative ring , possibly non-commutative, and let $p$ be a prime number. Let $E(A)$ be the ring of $p$-typical Witt vectors as constructed by Cuntz and Deninger and $W(A)$ be that constructed by Hesselholt. The goal of this paper is to answer the following question by Hesselholt: Is $HH_0(E(A)) $ isomorphic to $W(A)$? We show that in the case $p=2$, there is no such isomorphism possible if one insists it to be compatible with the Verscheibung operator and the Teichmuller map.
Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if for all nonunit elements $a,b,c in R$ such that $0 eq abc in I$, then either $ab in I$ or $c in I$. A number of results concerning weakly $1$-absorbing prime ideals and examples of weakly $1$-absorbing prime ideals are given. It is proved that if $I$ is a weakly $1$-absorbing prime ideal of a ring $R$ and $0 eq I_1I_2I_3 subseteq I$ for some ideals $I_1, I_2, I_3$ of $R$ such that $I$ is free triple-zero with respect to $I_1I_2I_3$, then $ I_1I_2 subseteq I$ or $I_3subseteq I$. Among other things, it is shown that if $I$ is a weakly $1$-absorbing prime ideal of $R$ that is not $1$-absorbing prime, then $I^3 = 0$. Moreover, weakly $1$-absorbing prime ideals of PIDs and Dedekind domains are characterized. Finally, we investigate commutative rings with the property that all proper ideals are weakly $1$-absorbing primes.
In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (= NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
