Matric generators of coalgebras and bialgebras

2019 ◽  
Vol 18 (08) ◽  
pp. 1950144
Author(s):  
Hiroshi Kihara
Keyword(s):  
Commutative Ring ◽  
Finite Subset ◽  
Finiteness Theorem ◽  
Finitely Generated ◽  
Semihereditary Ring ◽  
Local Finiteness

Takeuchi asserted that if a bialgebra [Formula: see text] over a field [Formula: see text] is finitely generated as a [Formula: see text]-algebra, then [Formula: see text] is a matric bialgebra. We introduce the notion of a matric coalgebra over a commutative ring [Formula: see text]. We show that if [Formula: see text] is faithfully projective as a [Formula: see text]-module, then [Formula: see text] is a matric coalgebra. Using this, we also show that if a bialgebra [Formula: see text] over a semihereditary ring [Formula: see text] is projective as a [Formula: see text]-module, then any finite subset of [Formula: see text] is contained in some matric subbialgebra. This result is a generalization of Takeuchi’s assertion and can be regarded as a local finiteness theorem on bialgebras.

Primary ideals with finitely generated radical in a commutative ring

1993 ◽  
Vol 78 (1) ◽  
pp. 201-221 ◽  
Author(s):  
Robert Gilmer ◽  
William Heinzer
Keyword(s):  
Commutative Ring ◽  
Finitely Generated ◽  
Primary Ideals

scholarly journals On a generalization of prime submodules of a module over a commutative ring

2017 ◽  
Vol 37 (1) ◽  
pp. 153-168
Author(s):  
Hosein Fazaeli Moghimi ◽  
Batool Zarei Jalal Abadi
Keyword(s):  
Commutative Ring ◽  
Dedekind Domain ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Maximal Ideals ◽  
Prime Submodules ◽  
Prime Submodule

‎Let $R$ be a commutative ring with identity‎, ‎and $n\geq 1$ an integer‎. ‎A proper submodule $N$ of an $R$-module $M$ is called‎ ‎an $n$-prime submodule if whenever $a_1 \cdots a_{n+1}m\in N$ for some non-units $a_1‎, ‎\ldots‎ , ‎a_{n+1}\in R$ and $m\in M$‎, ‎then $m\in N$ or there are $n$ of the $a_i$'s whose product is in $(N:M)$‎. ‎In this paper‎, ‎we study $n$-prime submodules as a generalization of prime submodules‎. ‎Among other results‎, ‎it is shown that if $M$ is a finitely generated faithful multiplication module over a Dedekind domain $R$‎, ‎then every $n$-prime submodule of $M$ has the form $m_1\cdots m_t M$ for some maximal ideals $m_1,\ldots,m_t$ of $R$ with $1\leq t\leq n$‎.

Weak Semiprojectivity in Purely Infinite Simple C*-Algebras

2007 ◽  
Vol 59 (2) ◽  
pp. 343-371 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Finite Subset ◽  
Finitely Generated ◽  
Linear Map ◽  
C Algebras ◽  
Direct Sum ◽  
Finitely Generated Groups ◽  
Positive Linear Map ◽  
Universal Coefficient Theorem

AbstractLet A be a separable amenable purely infinite simple C*-algebra which satisfies the Universal Coefficient Theorem. We prove that A is weakly semiprojective if and only if Ki(A) is a countable direct sum of finitely generated groups (i = 0, 1). Therefore, if A is such a C*-algebra, for any ε > 0 and any finite subset ℱ ⊂ A there exist δ > 0 and a finite subset ⊂ A satisfying the following: for any contractive positive linear map L : A → B (for any C*-algebra B) with ∥L(ab) – L(a)L(b)∥ < δ for a, b ∈ there exists a homomorphism h: A → B such that ∥h(a) – L(a)∥ < ε for a ∈ ℱ.

scholarly journals Inertial subalgebras of algebras possessing finite automorphism groups

1979 ◽  
Vol 28 (3) ◽  
pp. 335-345 ◽  
Author(s):  
Nicholas S. Ford
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Fixed Ring ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

Some properties of a family of quotients of the Rees algebra

2019 ◽  
Vol 18 (06) ◽  
pp. 1950113 ◽  
Author(s):  
Elham Tavasoli
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Proper Ideal ◽  
Rees Algebra ◽  
Finitely Generated ◽  
Flat Ideal

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a nonzero proper ideal of [Formula: see text]. In this paper, we study the properties of a family of rings [Formula: see text], with [Formula: see text], as quotients of the Rees algebra [Formula: see text], when [Formula: see text] is a semidualizing ideal of Noetherian ring [Formula: see text], and in the case that [Formula: see text] is a flat ideal of [Formula: see text]. In particular, for a Noetherian ring [Formula: see text], it is shown that if [Formula: see text] is a finitely generated [Formula: see text]-module, then [Formula: see text] is totally [Formula: see text]-reflexive as an [Formula: see text]-module if and only if [Formula: see text] is totally reflexive as an [Formula: see text]-module, provided that [Formula: see text] is a semidualizing ideal and [Formula: see text] is reducible in [Formula: see text]. In addition, it is proved that if [Formula: see text] is a nonzero flat ideal of [Formula: see text] and [Formula: see text] is reducible in [Formula: see text], then [Formula: see text], for any [Formula: see text]-module [Formula: see text].

Modules with annihilation property

2020 ◽  
pp. 2150126
Author(s):  
Rasul Mohammadi ◽  
Ahmad Moussavi ◽  
Masoome Zahiri
Keyword(s):  
Commutative Ring ◽  
Associative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Ordered Monoid ◽  
Zero Divisors ◽  
The Right ◽  
Totally Ordered Monoid

Let [Formula: see text] be an associative ring with identity. A right [Formula: see text]-module [Formula: see text] is said to have Property ([Formula: see text]), if each finitely generated ideal [Formula: see text] has a nonzero annihilator in [Formula: see text]. Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property ([Formula: see text]). We study and construct various classes of modules with Property ([Formula: see text]). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593–2601.], we introduce [Formula: see text]-dual McCoy modules and show that, for every strictly totally ordered monoid [Formula: see text], faithful symmetric modules are [Formula: see text]-dual McCoy. We then use this notion to give a characterization for modules with Property ([Formula: see text]). For a faithful symmetric right [Formula: see text]-module [Formula: see text] and a strictly totally ordered monoid [Formula: see text], it is proved that the right [Formula: see text]-module [Formula: see text] is primal if and only if [Formula: see text] is primal with Property ([Formula: see text]).

On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules

2020 ◽  
Vol 27 (1) ◽  
pp. 103-110
Author(s):  
Shahram Motmaen ◽  
Ahmad Yousefian Darani
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Finitely Generated ◽  
Module Homomorphism ◽  
Prime Submodule ◽  
Bezout Module

AbstractIn this paper, we introduce some classes of R-modules that are closely related to the classes of Prüfer, Dedekind and Bezout modules. Let R be a commutative ring with identity and set\mathbb{H}=\bigl{\{}M\mid M\text{ is an }R\text{-module and }\mathrm{Nil}(M)% \text{ is a divided prime submodule of }M\bigr{\}}.For an R-module {M\in\mathbb{H}}, set {T=(R\setminus Z(R))\cap(R\setminus Z(M))}, {\mathfrak{T}(M)=T^{-1}M} and {P=(\mathrm{Nil}(M):_{R}M)}. In this case, the mapping {\Phi:\mathfrak{T}(M)\to M_{P}} given by {\Phi(x/s)=x/s} is an R-module homomorphism. The restriction of Φ to M is also an R-module homomorphism from M into {M_{P}} given by {\Phi(x)=x/1} for every {x\in M}. A nonnil submodule N of M is said to be Φ-invertible if {\Phi(N)} is an invertible submodule of {\Phi(M)}. Moreover, M is called a Φ-Prüfer module if every finitely generated nonnil submodule of M is Φ-invertible. If every nonnil submodule of M is Φ-invertible, then we say that M is a Φ-Dedekind module. Furthermore, M is said to be a Φ-Bezout module if {\Phi(N)} is a principal ideal of {\Phi(M)} for every finitely generated submodule N of the R-module M. The paper is devoted to the study of the properties of Φ-Prüfer, Φ-Dedekind and Φ-Bezout R-modules.

On the compressed essential graph of a module over a commutative ring

2020 ◽  
pp. 2150125
Author(s):  
S. H. Payrovi ◽  
S. Babaei ◽  
E. Sengelen Sevim
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Essential Ideal

Let [Formula: see text] be a commutative ring and [Formula: see text] be an [Formula: see text]-module. The compressed essential graph of [Formula: see text], denoted by [Formula: see text] is a simple undirected graph associated to [Formula: see text] whose vertices are classes of torsion elements of [Formula: see text] and two distinct classes [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal of [Formula: see text]. In this paper, we study diameter and girth of [Formula: see text] and we characterize all modules for which the compressed essential graph is connected. Moreover, it is proved that [Formula: see text], whenever [Formula: see text] is Noetherian and [Formula: see text] is a finitely generated multiplication module with [Formula: see text].

Generators of Un(V) Over a Quasi Semilocal Semihereditary Ring

1981 ◽  
Vol 33 (5) ◽  
pp. 1232-1244 ◽  
Author(s):  
Hiroyuki Ishibashi
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Valuation Ring ◽  
Hermitian Form ◽  
Free Module ◽  
Fixed Element ◽  
Semihereditary Ring ◽  
Maximal Ideals ◽  
Sesquilinear Form

Let o be a quasi semilocal semihereditary ring, i.e., o is a commutative ring with 1 which has finitely many maximal ideals {Ai|i ∊ I} and the localization oAi at any maximal ideal Ai is a valuation ring. We assume 2 is a unit in o. Furthermore * denotes an involution on o with the property that there exists a unit θ in o such that θ* = –θ. V is an n-ary free module over o with f : V × V → o a λ-Hermitian form. Thus λ is a fixed element of o with λλ* = 1 and f is a sesquilinear form satisfying f(x, y)* = λf(y, x) for all x, y in V. Assume the form is nonsingular; that is, the mapping M → Hom (M, A) given by x → f( , x) is an isomorphism. In this paper we shall write f(x, y) = xy for x, y in V.

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