Here tensor products are defined for modules over a commutative ring $A$ with identity. I have only encountered tensor products in linear algebra(where the multilinearty of determinants naturally leads to the definition of tensor products.) But I didn’t have any idea how “powerful” this object will be. Recently I came up with a way to look at it when I am studying commutative algebra:
Let $A$ be a commutative ring with identity and $I$ an ideal of $A$. Given a left $A$-module $M$ we would “modify” $M$ so that it becomes a module over the quotient ring $A/I$. It turns out $M/IM$ is the module we are looking for, with the naturally induced $A/I$-module structure defined by $$(a+I)(m+IM)=am+IM.$$ This is useful when $I$ is a maximal ideal of $A$, because then we have a naturally induced vector space $M/IM$ over the field $A/I$. But using tensor products, we obtain another way to look at $M/IM$: $$(A/I)\otimes_A M\simeq M/IM.$$ In other words, the tensor product in this case is the “modification” we need.
As I understand, tensor products in this case simplify a lot when we study local rings and we only need to consider the algebraic properties of tensor products.
Are there any other perspectives to look at tensor products? (not limited to just algebra) At what other place will it be powerful?