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A Course in Ring Theory

Donald S. Passman

Chapter 18

Big Projectives - all with Video Answers

Educators

Chapter Questions

01:23

Problem 1

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

Linh Vu
Linh Vu
Numerade Educator
03:13

Problem 2

Suppose $P \cong \oplus \sum_{i=1}^{\infty} P_{i}$ is an infinite direct sum of countably generated projective $R$-modules $P_{i}$ with $T\left(P_{i}\right)=R .$ If $R$ is right Noetherian, show that there exists an epimorphism $\mu: P \rightarrow R$. Conclude from the Eilenberg Trick that $P$ is free.

Gideon Idumah
Gideon Idumah
Numerade Educator
03:56

Problem 3

Suppose $R$ is a right Noetherian ring and $P$ is a finitely generated projective $R$-module. Prove that $P$ is a progenerator if and only if $p^{\infty} \cong R^{\infty}$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
01:47

Problem 4

If $I$ is a right ideal of $R$, show that $T\left(I_{R}\right) \supseteq R I$. Furthermore, if $I=e R$ for some idempotent $e$, prove that $T(e R)=R e R$.

Srilakshmi E K
Srilakshmi E K
Numerade Educator
04:33

Problem 5

Find an example of a nonzero finitely generated projective $R$-module that is not a progenerator. If $R$ is a simple ring and $I$ is a nonzero right ideal of $R$, prove that $I_{R}$ is a generator.

Lucía Guerrero
Lucía Guerrero
Numerade Educator
08:25

Problem 6

Show that all nonzero $R$-modules are generators if and only if $R$ is a simple Artinian ring. If $R$ is assumed to be right Artinian, show that

Ely Crowder
Ely Crowder
Numerade Educator
01:26

Problem 7

Let $P$ be a projective $R$-module and say $P+Q=F=\cdot \sum_{i} f_{i} R$, where, of course, $F$ is free. If $x \in P$, write $x=\sum_{i} f_{i} \lambda_{i}(x)$ and apply the projection map $\pi_{P}$ to conclude that $P=P T(P)$. In particular, if $T(P) \neq R$, there exists a maximal ideal $M$ of $R$ with $P / P M=0$.

AG
Ankit Gupta
Numerade Educator
04:33

Problem 8

If $R$ is a Dedekind domain and $0 \neq A \triangleleft R$, show that $T\left(A_{R}\right)=A^{-1} A=$ $R$. Conclude that all nonzero finitely generated projective $R$-modules are progenerators.

Lucía Guerrero
Lucía Guerrero
Numerade Educator
03:13

Problem 9

Now assume that $R$ is a commutative domain and let $V$ be a torsion free $R$-module. If $T$ is a multiplicatively closed subset of $R$, verify all the steps in the construction of the module $V T^{-1}$. Furthermore, if $S \supseteq T$ is also a multiplicatively closed subset of $R$, prove that $\left(V T^{-1}\right) S^{-1} \cong V S^{-1}$ as $R S^{-1}$-modules.

Gideon Idumah
Gideon Idumah
Numerade Educator
03:56

Problem 10

Let $P$ be a prime ideal of the commutative integral domain $R$. If $R+P R_{P}=R_{P}$, prove that $P$ is a maximal ideal of $R$. In particular, conclude that Lemma $18.5$ fails for nonmaximal ideals.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator