In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
, a ring extension is a
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preservi ...
of
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
s, which makes an -
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
.
In this article, a ring extension of a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'' by an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
''I'' is a pair of a ring ''E'' and a surjective ring homomorphism
such that ''I'' is isomorphic (as an abelian group) to the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
of
In other words,
:
is a
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
of abelian groups. (This makes ''I'' a
two-sided ideal
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers pre ...
of ''E''.)
Given a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
''A'', an ''A''-extension is defined in the same way by replacing "ring" with "
algebra over ''A''" and "abelian groups" with "''A''-
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
".
An extension is said to be ''trivial'' if
splits; i.e.,
admits a
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...
that is an
algebra homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in ,
* F(kx) = kF ...
. This implies that ''E'' is isomorphic to the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of ''R'' and ''I''.
A
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
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
In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams.
The five lemma is not only valid for abelian categories but also w ...
, such a morphism is necessarily an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
, and so two extensions are equivalent if there is a morphism between them.
Examples
Example 1
Let's take the ring
of whole numbers and let's take the abelian group
(under addition) of binary numbers. Let ''E'' =
we can identify multiplication on ''E'' by
(where
is the homomorphism mapping even numbers to 0 and odd numbers to 1).
This gives the short exact sequence
:
Where ''p'' is the homomorphism mapping
.
Example 2
Let ''R'' be a commutative ring and ''M'' an ''R''-module. Let ''E'' = ''R'' ⊕ ''M'' be the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
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. We then have the short exact sequence
:
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 Nagata is a surname which can be either of Japanese (written: 永田 or 長田) or Fijian origin. Notable people with the surname include:
*Akira Nagata (born 1985), Japanese vocalist and actor
* Alipate Nagata, Fijian politician
*Anna Nagata (bor ...
calls this process the ''principle of idealization''.
References
*E. Sernesi:
Deformations of algebraic schemes'
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