202506111606

Tags : Module Theory

Examples of Modules

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Example

Every abelian group is is a module over the ring $\mathbb{Z}$. That too in exactly 1 way as there is only one morphism from $\mathbb{Z}$ to the the endomorphism ring of the group as $\mathbb{Z}$ is the intial ring.

Example

If $R$ is a field $k$, then we call the module a Vector space over the field $k$.

Example

Given any homomorphism of rings $\alpha :R\to S$, one can define a module with $\rho :R\times S\to S$ such that $\rho (r,s)=\alpha (r)\cdot s$. People usually just write this as $rs$ instead of $\alpha (r)s$.

Example

In the previous example, if $R$ is a commutative ring, and $\alpha$ maps $R$ to the center of $S$ then it would be called an $R$-algebra.

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References

• Modules