For questions about rings which are not necessarily commutative and modules over such rings.

In abstract algebra, the theory of rings which are not necessarily commutative is called "noncommutative algebra." In this way it is a generalization of commutative algebra. Some results from commutative algebra hold in noncommutative algebra, but many results break down.

The ring of quaternions was among the first motivating examples of noncommutative rings. Other familiar examples include the $n\times n$ matrix ring over any ring ($n>1$).

A few examples of some differences between commutative and noncommutative algebra:

  • If $R$ is a commutative ring, and $R^n\cong R^m$ as $R$ modules for some positive integers $m$ and $n$, then $m=n$. In contrast, there is a noncommutative ring such that $R^m\cong R^n$ for every pair of positive integers $m,n$.

  • Any commutative ring without zero divisors can be embedded in a field. There are examples of noncommutative rings without zero divisors which cannot be embedded into a division ring. This is one of many signs that show localization does not work well for many noncommutative rings.

  • The module $R_R$ may have different properties from the module $_RR$. For one thing, one could be Noetherian (or Artinian) without the other having the same property.

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