Varieties whose finitely generated members are free
نویسندگان
چکیده
منابع مشابه
Two Finitely Generated Varieties Having No Infinite Simple Members
Using a method of R. McKenzie, we construct a finitely generated semisimple variety of infinite type, and a finitely generated nonsemisimple variety of finite type, both having arbitrarily large finite but no infinite simple members. This amplifies M. Valeriote’s negative solution to Problem 11 from [1]. R. McKenzie [2] has constructed a finitely generated variety having arbitrarily large finit...
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A finitely generated $R$-module is said to be a module of type ($F_r$) if its $(r-1)$-th Fitting ideal is the zero ideal and its $r$-th Fitting ideal is a regular ideal. Let $R$ be a commutative ring and $N$ be a submodule of $R^n$ which is generated by columns of a matrix $A=(a_{ij})$ with $a_{ij}in R$ for all $1leq ileq n$, $jin Lambda$, where $Lambda $ is a (possibly infinite) index set. ...
متن کاملMULTIPLICATION MODULES THAT ARE FINITELY GENERATED
Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. An $R$-module $M$ is called a multiplication module if for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N = IM$. It is shown that over a Noetherian domain $R$ with dim$(R)leq 1$, multiplication modules are precisely cyclic or isomorphic to an invertible ideal of $R$. Moreover, we give a charac...
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Let A be a k-element algebra whose chief factor size is c. We show that if B is in the variety generated by A, then any abelian chief factor of B that is not strongly abelian has size at most ck−1. This solves Problem 5 of The Structure of Finite Algebras, by D. Hobby and R. McKenzie. We refine this bound to c in the situation where the variety generated by A omits type 1. As a generalization, ...
متن کاملFree modules, finitely-generated modules
The following definition is an example of defining things by mapping properties, that is, by the way the object relates to other objects, rather than by internal structure. The first proposition, which says that there is at most one such thing, is typical, as is its proof. Let R be a commutative ring with 1. Let S be a set. A free R-moduleM on generators S is an R-module M and a set map i : S →...
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ژورنال
عنوان ژورنال: Algebra universalis
سال: 2018
ISSN: 0002-5240,1420-8911
DOI: 10.1007/s00012-018-0480-5