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steenis
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If the moderators agree, I have made a thread on the book of Bland - Rings and their Modules 2011.
Hi Steenis,steenis said:I want to take you to Example 1 of section 1.5 p.33: "$\mbox{Hom} _R (M,N)$ as a left R-module.
Let $M$ and $N$ be two right R-modules. R is not necessarely commutative.
On 9th line of this example, Bland claims that $\mbox{Hom} _R (M,N)$ is a left R-module if we define $[r\bullet f](x)=f(xr)$ for $r\in R$ and $x\in M$. To prove that $\mbox{Hom} _R (M,N)$ is a left R-module, we have to prove conditions (1), (2), (3) and (4) of Definition 1.4.1 p.26 (adapted for left R-modules). Bland proves (3) as follows, $r\in R$, $s\in R$, $x\in M$:
$[s\bullet (r\bullet f)[(x)$ = $[r\bullet f](xs)$ = $[f]((xs)r)$ = $[f](x(sr))$ = $[(sr)\bullet f](x)$
To prove conditions (1), (2), and (4) is easy.
However, he omits to prove that if $f$ is an R-map then $[r\bullet f]$ is an R-map for $r\in R$, i.e., if $f\in \mbox{Hom} _R (M,N)$ then $[r\bullet f] \in \mbox{Hom} _R (M,N)$.
And that is my problem.
Given for $r\in R$, $a\in R$, and $x\in M$ is $[r\bullet f](x)=f(xr)$ and $f(xa)=f(x)a$, we have to prove that $[r\bullet f](xa)$ = $[r\bullet f](x)a$. Here is a start:
$[r\bullet f](xa)$ = $[f]((xa)r)$ = $[f](x(ar))$ = ? = $[r\bullet f](x)a$
Who can fill in the missing steps ?
Euge said:Hi steenis,
If the given conditions are correct, then there is problem with Bland's claim here. If $R$ is commutative, then indeed $\operatorname{Hom}_R(M,N)$ is a left $R$-module with the given $R$-action. However, if $R$ is non-commutative, then the result need not hold. For suppose $R$ is a non-commutative, unital ring. We may view $R$ as a right $R$-module in the usual way. Consider the element $i \in \operatorname{Hom}_R(R,R)$ given by $i(r) = r$. Then $[b\cdot i](a) = i(ab) = ab$, but $[b\cdot i](1)a = i(1b)a = i(b)a = ba\neq ab = [b\cdot i](a)$. Hence, $[b\cdot i]\notin \operatorname{Hom}_R(R,R)$.
By the way, I've noticed that you and Peter are working on issues between Bland's and Rotman's texts. I will address some of those questions if I have time.
The purpose of this book is to provide a comprehensive and in-depth discussion of rings and their modules, which are important mathematical structures used in various fields of science and engineering.
This book is intended for graduate students and researchers in mathematics and related fields who have a basic understanding of abstract algebra and are interested in learning more about rings and their modules.
This book covers a wide range of topics related to rings and their modules, including basic definitions and properties, homomorphisms and isomorphisms, submodules, quotient modules, direct sums and products, and modules over principal ideal domains.
It is recommended that readers have a solid understanding of abstract algebra, including groups, rings, and fields, as well as basic knowledge of linear algebra and vector spaces. Some familiarity with basic set theory and logic may also be helpful.
Yes, this book includes various examples and applications of rings and modules in fields such as number theory, algebraic geometry, and coding theory. It also discusses the connections between rings and modules and other mathematical structures, such as vector spaces and algebras.