Question

Prove that $V$ is a generator if and only if $T(V)=R$.

   Prove that $V$ is a generator if and only if $T(V)=R$.

A Course in Ring Theory

A Course in Ring Theory

Donald S. Passman 2004 Edition

Chapter 17, Problem 1

Instant Answer

Solved by Expert Linh Vu

10/16/2020

Step 1

First, we need to show that if $V$ is a generator, then $T(V) = R$. Assume that $V$ is a generator. This means that for any vector $r \in R$, there exists a vector $v \in V$ such that $T(v) = r$. Since $T$ is a linear transformation, we have $T(V) = \{T(v) \mid v Show more…

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Prove that $V$ is a generator if and only if $T(V)=R$.

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Key Concepts

Module Homomorphisms

Module homomorphisms are maps between modules that preserve the module structure, and they play a key role in the construction of the trace T(V). By considering all R-linear maps from V to R, one aggregates their images to form the trace. These maps are fundamental in assessing whether every element of R can be obtained from some image of V, which is a crucial step in proving that V is a generator if and only if T(V) equals R.

Generator

In module theory, a generator is an object (often a module) such that every other module in the category is a quotient of a direct sum of copies of it. This means that the module is sufficiently 'rich' to produce, via homomorphisms and direct sums, any module in the category. The concept is crucial in understanding how the structure of the category can be built from a single module and is a central idea in Morita theory.

Trace of a Module

The trace of a module V in a ring R, often denoted by T(V), is defined as the ideal of R generated by the images of all R-linear maps from V to R. In other words, it is the sum of the submodules (ideals) obtained from applying every possible homomorphism from V into R. This concept is used to measure the extent to which V 'covers' R via its morphisms, linking directly to the idea of V being a generator when T(V) equals the whole ring R.

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