Ring extension

In commutative algebra, a ring extension is a ring homomorphism $R\to S$ of commutative rings, which makes an -algebra.

In this article, a ring extension of a ring ''R'' by an abelian group ''I'' is a pair of a ring ''E'' and a surjective ring homomorphism $\phi:E\to R$ such that ''I'' is isomorphic (as an abelian group) to the kernel of $\phi.$ In other words,
:$0 \to I \to E \overset R \to 0$
is a short exact sequence of abelian groups. (This makes ''I'' a two-sided ideal of ''E''.)

Given a commutative ring ''A'', an ''A''-extension 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'' if $\phi$ splits; i.e., $\phi$ admits a section that is an algebra homomorphism. This implies that ''E'' is isomorphic to the direct product of ''R'' and ''I''.

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.

Examples

Example 1

Let's take the ring $\mathbb Z$ of whole numbers and let's take the abelian group $\mathbb Z_2$(under addition) of binary numbers. Let ''E'' = $\mathbb Z \oplus \mathbb Z_2$ we can identify multiplication on ''E'' by $(x,a) \cdot (y,b) = (xy,\phi(x)a+\phi(y)b)$(where $\phi:\mathbb Z \to \mathbb Z_2$ is the homomorphism mapping even numbers to 0 and odd numbers to 1).
This gives the short exact sequence
:$0\to \mathbb Z \to E \overset \mathbb Z \to 0$
Where ''p'' is the homomorphism mapping $(x,a) \mapsto a\phi(x)$.

Example 2

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. 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''. 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''.

References

*E. Sernesi:
Deformations of algebraic schemes
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Ring theory