Big list of undergraduate exercises in module theory

Question

I would like to write down a big list of exercises in module theory addressed to a course of introduction to this subject. The background of the average student will be the definition of ring, and some basic feeling with fields, PIDs, UFDs, domains in general and linear algebra.

I am looking for both moral-exercises (an example in linear algebra would be "is it true that given $v_1,v_2,v_3$ such that any pair is linearly independent, then they are linearly independent?) and puzzle-exercises (an example would in linear algebra would be "prove that any endomorphism of $\mathbb{R}^3$ has an invariant hyperplane").

The line of the course is far from computational algebra and is closer to a survey on abelian categories (without saying abelian category).

Answer 1

• A surjective endomorphism $M \to M$ of finitely generated module is in fact injective.

Answer 2

• Let $R$ be a unitary ring. If there are two different left inverses to an element $x\in R$ then there is an infinite number of left inverses for $x$. (extremely instructive, zero technology; a smart student with only the definition of ring can do it)
• the center of a simple ring is a field (one liner if you have the right idea)
• Let $M$ be an $R$-module and let $f, g \in End(M)$ such that if $f = fgf$ then $M\cong \ker f \oplus M'$ and $M\cong \text{im }f\oplus M''$, i.e. $\ker f$ and $\text{im } f$ are direct summands of $M$ (Bonus: Prove that the converse is also true.)
• If $M$ is either artinian or noetherian, then it has the IBN.
• Is there a simple ring which is neither right noetherian, right artinian, left noetherian nor left artinian?
• Let $P_R$ be a right $R$-module. Define the trace $\textbf{tr} (M)$ of $P$ in a module $M_R$ to be the sum $$ \textbf{tr} (M) = \sum_{f : P \to M} \text{im } f $$ Prove that:
  1. $\textbf{tr} (M) = \text{im }\varphi$ for some $R$-module morphism $\varphi : P^{(I)} \to M$ ;
  2. $M\mapsto \textbf{tr} (M)$ is a functor.
• If I were you I'd package the whole injective-flat-projective story into some hidden definitions on orthogonality and exercises thereof.
• Same story for the tensor product stuff, we're in 2017: first a universal property and then you build a concrete object; and you use the universal property to do all the classical exercises.