Difference between linear space theory and $R$ -module theory

Question

Let $R$ be a commutative ring.

What are propositions about $R$-modules that do not generally hold but are true when $R$ is a field?

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So-called "fundamental theorem of linear algebra"

$$(\mathrm{Im} f)^\perp=\mathrm{Ker} f^*$$

is not a feature of linear spaces, because for any $R$-homomorphism $f:M\to N$ between $R$-modules,

$$\begin{align} \varphi \in (\mathrm{Im} f)^\perp &\Leftrightarrow \forall n \in \mathrm{Im}f. \varphi(n)=0\\ &\Leftrightarrow \forall m\in M. \varphi(f(m))=0\\ &\Leftrightarrow f^*(\varphi)=0\\ &\Leftrightarrow \varphi\in \mathrm{Ker}f^*. \end{align}$$

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I know only two examples.

1. Basis theorem. Every linear space has a basis, but not all modules.
2. If $a_1, \dots, a_m$ are linearly dependent, one of them is a linear combination of others.

(linear space theory) - (module theory) = (dimension theory)?

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If $a_1, \dots, a_m\in V$ are linearly independent and $b_1, \dots, b_n \in V$ spans $V$, $m\leq n$ holds. However, this is also true for modules over a commutative ring(link).

Many properties of linear spaces can be generalized to $R$-modules. I want to know counterexamples.

Answer 1

The "most important" fact about vector spaces which fails for modules in general is exactly what you noted: every vector space has a basis, but not every module has a basis. Indeed, the following are equivalent (for the rest of this post, $R$ will be a commutative unital ring):

• Every $R$-module is free (i.e. every $R$-module has a basis)
• If $a_1, \dots, a_n$ are linearly dependent elements of an $R$-module $M$, then one of them is a linear combination of the others.
• $R$ is a field (i.e. $R$-modules are vector spaces)

Another commonly-used fact in linear algebra is that every subspace has a complementary subspace. Rings for which this is true of all submodules need not be fields, but they are nicely classified. Precisely, the following are equivalent:

• If $M$ is an $R$-module with a submodule $N$, then there is some submodule $N'$ of $M$ such that $M = N \oplus N'$.
• $R$ is isomorphic (as a ring) to a direct product of finitely many fields.
• $R$ (viewed as a module over itself) is a direct sum of simple $R$-modules.
• Every $R$-module is a direct sum of simple $R$-modules.
• $R$ is Artinian and reduced.

For many properties (in the language of modules) we have a good understanding of the rings $R$ for which the property holds for all $R$-modules. Do you have any specific properties you're curious about? It's a bit silly to ask for examples of properties of vector spaces and modules for which those properties don't hold: there are simply too many to list.