Algebra extension

{{Short description|Surjective ring homomorphism with a given codomain}}

{{hat note|For the ring-theoretic equivalent of a field extension, see Subring#Ring 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, $\phi$ ) consisting of a ring E and a ring homomorphism $\phi$ that fits into the short exact sequence of abelian groups:

: $0 \to I \to E \overset{\phi}{{}\to{}} R \to 0.$ {{harvnb|Sernesi|2007|loc=1.1.1.}}

This makes I 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 $\phi$ splits; i.e., $\phi$ admits a section that is a ring homomorphismTypical 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. (see {{section link||Example: trivial extension}}).

A morphism between extensions of R by I, over say A, is an algebra homomorphism E → E{{'}} 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.

Trivial extension example

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

: $(a, x) \cdot (b, y) = (ab, ay + bx).$

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 $\operatorname{Sym}(M)/\bigoplus_{n \ge 2} \operatorname{Sym}^n(M)$ where $\operatorname{Sym}(M)$ is the symmetric algebra of M.{{cite journal |last1=Anderson |first1=D. D. |last2=Winders |first2=M. |title=Idealization of a Module |journal=Journal of Commutative Algebra |date=March 2009 |volume=1 |issue=1 |pages=3–56 |doi=10.1216/JCA-2009-1-1-3 |s2cid=120720674 |url=https://projecteuclid.org/journals/journal-of-commutative-algebra/volume-1/issue-1/Idealization-of-a-Module/10.1216/JCA-2009-1-1-3.full |issn=1939-2346|doi-access=free }} We then have the short exact sequence

: $0 \to M \to E \overset{p}{{}\to{}} R \to 0$

where p is the projection. Hence, E is an extension of R by M. It is trivial since $r \mapsto (r, 0)$ is a section (note this section is a ring homomorphism since $(1, 0)$ is the multiplicative identity of E). Conversely, every trivial extension E of R by I is isomorphic to $R \oplus I$ if $I^2 = 0$ . Indeed, identifying $R$ as a subring of E using a section, we have $(E, \phi) \simeq (R \oplus I, p)$ via $e \mapsto (\phi(e), e - \phi(e))$ .

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.{{citation|mr=0155856|last=Nagata|first=Masayoshi|author-link=Masayoshi Nagata|title=Local Rings|series=Interscience Tracts in Pure and Applied Mathematics|volume=13|publisher=Interscience Publishers a division of John Wiley & Sons|place=New York-London|year=1962|isbn=0-88275-228-6}}

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 $R/I$ -bimodule.

More generally, an extension by a nilpotent ideal is called a nilpotent extension. For example, the quotient $R \to R_{\mathrm{red}}$ of a Noetherian commutative ring by the nilradical is a nilpotent extension.

In general,

: $0 \to I^n/I^{n-1} \to R/I^{n-1} \to R/I^n \to 0$

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.

References

{{cite book |last1=Sernesi |first1=Edoardo |title=Deformations of Algebraic Schemes |date=20 April 2007 |publisher=Springer Science & Business Media |isbn=978-3-540-30615-3 |url=https://books.google.com/books?id=xkcpQo9tBN8C |language=en}}

Algebra extension

Trivial extension example

Square-zero extension

See also

References

Further reading