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تسجيل مستخدم جديدPrime Ideals of q-Commutative Power Series Rings
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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).
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We continue the first and second authors study of $q$-commutative power series rings $R=k_q[[x_1,ldots,x_n]]$ and Laurent series rings $L=k_q[[x^{pm 1}_1,ldots,x^{pm 1}_n]]$, specializing to the case in which the commutation parameters $q_{ij}$ are a
ll roots of unity. In this setting, $R$ is a PI algebra, and we can apply results of De Concini, Kac, and Procesi to show that $L$ is an Azumaya algebra whose degree can be inferred from the $q_{ij}$. Our main result establishes an exact criterion (dependent on the $q_{ij}$) for determining when the centers of $L$ and $R$ are commutative Laurent series and commutative power series rings, respectively. In the event this criterion is satisfied, it follows that $L$ is a unique factorization ring in the sense of Chatters and Jordan, and it further follows, by results of Dumas, Launois, Lenagan, and Rigal, that $R$ is a unique factorization ring. We thus produce new examples of complete, local, noetherian, noncommutative, unique factorization rings (that are PI domains).
We study prime ideals in skew power series rings $T:=R[[y;tau,delta]]$, for suitably conditioned right noetherian complete semilocal rings $R$, automorphisms $tau$ of $R$, and $tau$-derivations $delta$ of $R$. These rings were introduced by Venjakob,
motivated by issues in noncommutative Iwasawa theory. Our main results concern Cutting Down and Lying Over. In particular, under the additional assumption that $delta = tau - id$ (a basic feature of the Iwasawa-theoretic context), we prove: If $I$ is an ideal of $R$, then there exists a prime ideal $P$ of $S$ contracting to $I$ if and only if $I$ is a $delta$-stable $tau$-prime ideal of $R$. Our approach essentially depends on two key ingredients: First, the algebras considered are zariskian (in the sense of Li and Van Oystaeyen), and so the ideals are all topologically closed. Second, topological arguments can be used to apply previous results of Goodearl and the author on skew polynomial rings.
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 nonu
nit 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.
A definition of quasi-flat left module is proposed and it is shown that any left module which is either quasi-projective or flat is quasi-flat. A characterization of local commutative rings for which each ideal is quasi-flat (resp. quasi-projective)
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An important instance of Rota-Baxter algebras from their quantum field theory application is the ring of Laurent series with a suitable projection. We view the ring of Laurent series as a special case of generalized power series rings with exponents
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