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'', $\phi$) consisting of a ring ''E'' and a ring homomorphism $\phi$ that fits into the short exact sequence of abelian groups:
:$0 \to I \to E \overset R \to 0.$
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 homomorphism (see ).

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 $\operatorname(M)/\bigoplus_ \operatorname^n(M)$ where $\operatorname(M)$ is the symmetric algebra of ''M''. We then have the short exact sequence
:$0 \to M \to E \overset R \to 0$
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''.

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 $R \to R_$ of a Noetherian commutative ring by the nilradical is a nilpotent extension.

In general,
:$0 \to I^n/I^ \to R/I^ \to R/I^n \to 0$
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.

See also

* Formally smooth map
*The Wedderburn principal theorem, a statement about an extension by the Jacobson radical.

References

*

Further reading

algebra extension at nLabinfinitesimal extension at nLabExtension of an associative algebra at Encyclopedia of Mathematics

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Ring theory