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.

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 Levi-Civita connections for a class of noncommutative minimal surfaces

2021 ◽  
pp. 2150194
Author(s):  
Joakim Arnlind
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Hermitian Form ◽  
Projective Modules ◽  
Fuzzy Sphere ◽  
Finitely Generated ◽  
Natural Concept ◽  
The Matrix ◽  
Pseudo Inverse ◽  
Torsion Free

In this paper, we study connections on hermitian modules, and show that metric connections exist on regular hermitian modules; i.e. finitely generated projective modules together with a non-singular hermitian form. In addition, we develop an index calculus for such modules, and provide a characterization in terms of the existence of a pseudo-inverse of the matrix representing the hermitian form with respect to a set of generators. As a first illustration of the above concepts, we find metric connections on the fuzzy sphere. Finally, the framework is applied to a class of noncommutative minimal surfaces, for which there is a natural concept of torsion, and we prove that there exist metric and torsion free connections for every minimal surface in this class.

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.

scholarly journals Determinants in Projective Modules

1961 ◽  
Vol 18 ◽  
pp. 27-36 ◽  
Author(s):  
Oscar Goldman
Keyword(s):  
Free Module ◽  
Exterior Algebra ◽  
Homogeneous Component ◽  
Projective Modules ◽  
Finitely Generated ◽  
Rank One ◽  
Definition Of

The definition of the determinant of an endomorphism of a free module depends on the following fact: If F is a free R-module of rank n, then the homogeneous component ∧nF, of degree n, of the exterior algebra ∧ F of F is a free R-module of rank one. If a is an endomorphism of F, then a extends to an endomorphism of ∧ F which in ∧nF is therefore multiplication by an element of R. That factor is then defined to be the determinant of α. (A discussion of this theory may be found in [4].)This procedure cannot be applied in general to finitely generated projective modules since, for such modules, it may happen that no homogeneous component of the exterior algebra is free of rank one.

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

Inverse complements and strongly unit regular elements

2021 ◽  
pp. 2250190
Author(s):  
Umashankara Kelathaya ◽  
Savitha Varkady ◽  
Manjunatha Prasad Karantha
Keyword(s):  
Partial Order ◽  
Regular Element ◽  
Generalized Inverse ◽  
Generalized Inverses ◽  
Order Unit ◽  
Regular Elements ◽  
Strongly Regular ◽  
Minus Partial Order

In this paper, the notion of “strongly unit regular element”, for which every reflexive generalized inverse is associated with an inverse complement, is introduced. Noting that every strongly unit regular element is unit regular, some characterizations of unit regular elements are obtained in terms of inverse complements and with the help of minus partial order. Unit generalized inverses of given unit regular element are characterized as sum of reflexive generalized inverses and the generators of its annihilators. Surprisingly, it has been observed that the class of strongly regular elements and unit regular elements are the same. Also, several classes of generalized inverses are characterized in terms of inverse complements.

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 $1\leq t\leq n$‎.

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