On a Class of Projective Modules Over Central Separable Algebras

1971 ◽  
Vol 14 (3) ◽  
pp. 415-417 ◽  
Author(s):  
George Szeto
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Projective Module ◽  
Projective Modules ◽  
Finitely Generated ◽  
Perfect Field ◽  
Maximal Ideals ◽  
Ring Of Polynomials ◽  
Separable Algebras

In [5], DeMeyer extended one consequence of Wedderburn's theorem; that is, if R is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents except 0 and 1 or if R is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective module over a central separable R-algebra A.

Projective Modules Over Central Separable Algebras

1969 ◽  
Vol 21 ◽  
pp. 39-43 ◽  
Author(s):  
F. R. DeMeyer
Keyword(s):  
Finite Number ◽  
Commutative Rings ◽  
Projective Modules ◽  
Principal Ideal Domain ◽  
Ideal Structure ◽  
Perfect Field ◽  
Maximal Ideals ◽  
Ring Of Polynomials ◽  
Separable Algebras ◽  
The Ideal

In (2), M. Auslander and O. Goldman laid the foundations for the study of central separable algebras. For unexplained terminology and notation, see (2). Here we are interested in projective modules and the ideal structure of a central separable algebra A over some special commutative rings K. When K is a field, one consequence of Wedderburn's Theorem is that there is a unique (up to isomorphism) irreducible A-module. We show here that if K is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents other than 0 and 1 or if K is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective A-module. An example in (3) shows that this result fails if one only assumes that K is a principal ideal domain.

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 $1\leq t\leq n$‎.

scholarly journals Rank Element of a Projective Module

1965 ◽  
Vol 25 ◽  
pp. 113-120 ◽  
Author(s):  
Akira Hattori
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Local Ring ◽  
Projective Module ◽  
Projective Modules ◽  
Local Theorem ◽  
Complete Local Ring ◽  
A Finite Group

In § 1 of this note we first define the trace of an endomorphism of a projective module P over a non-commutative ring A. Then we call the trace of the identity the rank element r(P) of P, which we shall illustrate by several examples. For a projective module P over the groupalgebra of a finite group G, the rank element of P is essentially the character of G in P. In § 2 we prove that under certain assumption two projective modules Pi and P2 over an algebra over a complete local ring o are isomorphic if and only if their rank elements are identical. This is a type of proposition asserting that two representations are equivalent if and only if their characters are identical, and in fact, when A is the groupalgebra, the above theorem may be considered as another formulation of Swan’s local theorem [9]).

scholarly journals Polynomial rings and their projective modules

1989 ◽  
Vol 113 ◽  
pp. 121-128 ◽  
Author(s):  
Dorin Popescu
Keyword(s):  
Noetherian Ring ◽  
Projective Module ◽  
Polynomial Rings ◽  
Projective Modules ◽  
Central Question ◽  
Finitely Generated

Let R be a regular noetherian ring. A central question concerning projective modules over polynomial R-algebras is the following.(1.1) BASS-QUILLEN CONJECTURE ([2] Problem IX, [10]). Every finitely generated projective module P over a polynomial R-algebra R[T], T = (T1,…, Tn) is extended from R, i.e.P≊R[T]⊗R P/(T)P.

PROJECTIVE MODULES AND DIVISOR HOMOMORPHISMS

2003 ◽  
Vol 02 (04) ◽  
pp. 435-449 ◽  
Author(s):  
ALBERTO FACCHINI ◽  
FRANZ HALTER-KOCH
Keyword(s):  
Commutative Ring ◽  
Projective Modules ◽  
Finitely Generated ◽  
Commutative Monoids ◽  
Isomorphism Classes

We study some applications of the theory of commutative monoids to the monoid [Formula: see text] of all isomorphism classes of finitely generated projective right modules over a (not necessarily commutative) ring R.

scholarly journals Clean coalgebras and clean comodules of finitely generated projective modules

2021 ◽  
Vol 31 (2) ◽  
pp. 251-260
Author(s):  
N. P. Puspita ◽  
◽  
I. E. Wijayanti ◽  
B. Surodjo ◽  
◽  
...  
Keyword(s):  
Tensor Product ◽  
Commutative Ring ◽  
Sufficient Conditions ◽  
Projective Modules ◽  
Morita Context ◽  
Finitely Generated ◽  
Module Homomorphism

Let R be a commutative ring with multiplicative identity and P is a finitely generated projective R-module. If P∗ is the set of R-module homomorphism from P to R, then the tensor product P∗⊗RP can be considered as an R-coalgebra. Furthermore, P and P∗ is a comodule over coalgebra P∗⊗RP. Using the Morita context, this paper give sufficient conditions of clean coalgebra P∗⊗RP and clean P∗⊗RP-comodule P and P∗. These sufficient conditions are determined by the conditions of module P and ring R.

scholarly journals Decompositions of modules into projective modules and CS-modules

2000 ◽  
Vol 62 (1) ◽  
pp. 159-164
Author(s):  
Somyot Plubtieng
Keyword(s):  
Projective Module ◽  
Injective Module ◽  
Projective Modules ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Noetherian Module ◽  
Direct Sum ◽  
Continuous Module

Let M be a right R-module. It is shown that M is a locally Noetherian module if every finitely generated module in σ[M] is a direct sum of a projective module and a CS-module. Moreover, if every module in σ[M] is a direct sum of a projective module and a CS-module, then every module in σ[M] is a direct sum of modules which are either indecomposable projective or uniform Σ-quasi-injective. In particular, if every module in σ[M] is a direct sum of a projective module and a quasi-continuous module, then every module in σ[M] is a direct sum of a projective module and a quasi-injective module.

Rank and dimension functions

2015 ◽  
Vol 29 ◽  
pp. 144-155
Author(s):  
K. Prasad ◽  
Nupur Nandini ◽  
Divya Shenoy
Keyword(s):  
Vector Space ◽  
Partial Order ◽  
Commutative Ring ◽  
Free Module ◽  
Projective Modules ◽  
Finitely Generated ◽  
The Matrix ◽  
Dimension Functions ◽  
Rank Of A Matrix ◽  
Minus Partial Order

In this paper, we invoke theory of generalized inverses and minus partial order on regular matrices over a commutative ring to define rank–function for regular matrices and dimension–function for finitely generated projective modules which are direct summands of a free module. Some properties held by the rank of a matrix and the dimension of a vector space over a field are generalized. Also, a generalization of rank-nullity theorem has been established when the matrix given is regular.

scholarly journals On Trace for Modules

1970 ◽  
Vol 40 ◽  
pp. 121-131 ◽  
Author(s):  
Howard B. Beckwith
Keyword(s):  
Characteristic Zero ◽  
Projective Module ◽  
Canonical Homomorphism ◽  
Projective Modules ◽  
Finitely Generated ◽  
Group Characters ◽  
Classical Mathematics ◽  
Square Matrix

Classically, trace was defined as the sum of the diagonal entries of a square matrix with entries in a field. This notion played an important role in classical mathematics, e.g. in the theory of algebras over a field of characteristic zero, and in the theory of group characters (as in [1]). A generalization to endomorphisms of a finitely generated projective module over any ring R with unit is well-known. For such a module P the canonical homomorphism Ψ: P* ⊗ P→ EndR(P) is an isomorphism. Then the composite ε˚Ψ-1: EndR(P) → P* ↗ P→ R, where e denotes “evaluation”, is a homomorphism which coincides with the classical trace whenever P is free. This version of trace has been used by Hattori [3] and others to study projective modules. However, this approach to trace is limited to the finitely generated projective modules, since it can be shown that Ψ is an isomorphism if and only if P is finitely generated and projective.

scholarly journals QF-3 endomorphism rings of Σ-quasi-projective modules

1988 ◽  
Vol 30 (2) ◽  
pp. 215-220 ◽  
Author(s):  
José L. Gómez Pardo ◽  
Nieves Rodríguez González
Keyword(s):  
Endomorphism Ring ◽  
Projective Module ◽  
Sufficient Conditions ◽  
Torsion Theory ◽  
Projective Modules ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Quotient Module ◽  
Qf Ring ◽  
Necessary And Sufficient

A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a Σ-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and σ[M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End(RM) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of σ[M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].

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