Algebra extension

In abstract algebra, an algebra extension is the ring-theoretic equivalent of a group extension.

Precisely, a ring extension of a ring R by an abelian group I is a pair (E, ) consisting of a ring E and a ring homomorphism that fits into the short exact sequence of abelian groups:

[1]

Note I is then isomorphic to a two-sided ideal of E. Given a commutative ring A, an A-extension or an extension of an A-algebra is defined in the same way by replacing "ring" with "algebra over A" and "abelian groups" with "A-modules".

An extension is said to be trivial or to split if splits; i.e., admits a section that is a ring homomorphism.[2] (see § Example: trivial extension).

A morphism between extensions of R by I, over say A, is an algebra homomorphism EE' that induces the identities on I and R. By the five lemma, such a morphism is necessarily an isomorphism, and so two extensions are equivalent if there is a morphism between them.

Example: trivial extension

Let R be a commutative ring and M an R-module. Let E = RM be the direct sum of abelian groups. Define the multiplication on E by

Note that identifying (a, x) with a + εx where ε squares to zero and expanding out (a + εx)(b + εy) yields the above formula; in particular we see that E is a ring. It is sometimes called the algebra of dual numbers. Alternatively, E can be defined as where is the symmetric algebra of M.[3] We then have the short exact sequence

where p is the projection. Hence, E is an extension of R by M. It is trivial since is a section (note this section is a ring homomorphism since is the multiplicative identity of E). Conversely, every trivial extension E of R by I is isomorphic to if . Indeed, identifying as a subring of E using a section, we have via .[1]

One interesting feature of this construction is that the module M becomes an ideal of some new ring. In his book Local Rings, Nagata calls this process the principle of idealization.[4]

Square-zero extension

Especially in deformation theory, it is common to consider an extension R of a ring (commutative or not) by an ideal whose square is zero. Such an extension is called a square-zero extension, a square extension or just an extension. For a square-zero ideal I, since I is contained in the left and right annihilators of itself, I is a -bimodule.

More generally, an extension by a nilpotent ideal is called a nilpotent extension. For example, the quotient of a Noetherian commutative ring by the nilradical is a nilpotent extension.

In general,

is a square-zero extension. Thus, a nilpotent extension breaks up into successive square-zero extensions. Because of this, it is usually enough to study square-zero extensions in order to understand nilpotent extensions.

See also

References

  1. Sernesi 2007, 1.1.1.
  2. Typical references require sections be homomorphisms without elaborating whether 1 is preserved. But since we need to be able to identify R as a subring of E (see the trivial extension example), it seems 1 needs to be preserved.
  3. Anderson, D. D.; Winders, M. (March 2009). "Idealization of a Module". Journal of Commutative Algebra. 1 (1): 3–56. doi:10.1216/JCA-2009-1-1-3. ISSN 1939-2346. S2CID 120720674.
  4. Nagata, Masayoshi (1962), Local Rings, Interscience Tracts in Pure and Applied Mathematics, vol. 13, New York-London: Interscience Publishers a division of John Wiley & Sons, ISBN 0-88275-228-6, MR 0155856

Further reading

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