In
ring theory
In algebra, ring theory is the study of rings— algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their re ...
, a Peirce decomposition is a decomposition of an algebra as a sum of
eigenspace
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of commuting
idempotent element
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
s.
The Peirce decomposition for
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
s was introduced by . A similar but more complicated Peirce decomposition for
Jordan algebra
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:
# xy = yx (commutative law)
# (xy)(xx) = x(y(xx)) ().
The product of two elements ''x'' and ''y'' in a Jordan al ...
s was introduced by .
Peirce decomposition for associative algebras
If ''e'' is a commuting idempotent (''e''
2 = ''e'' and ''e'' is in the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
of ''A'') in an associative algebra ''A'', then the two-sided Peirce decomposition writes ''A'' as the direct sum of ''eAe'', ''eA''(1 − ''e''), (1 − ''e'')''Ae'', and (1 − ''e'')''A''(1 − ''e''). There are also left and right Peirce decompositions, where the left decomposition writes ''A'' as the direct sum of ''eA'' and (1 − ''e'')''A'', and the right one writes ''A'' as the direct sum of ''Ae'' and ''A''(1 − ''e'').
More generally, if ''e''
1, ..., ''e''
''n'' are mutually orthogonal idempotents with sum 1, then ''A'' is the direct sum of the spaces ''e''
''i''''Ae''
''j'' for 1 ≤ ''i'', ''j'' ≤ ''n''.
Blocks
An idempotent of a ring is called central if it commutes with all elements of the ring.
Two idempotents ''e'', ''f'' are called orthogonal if ''ef'' = ''fe'' = 0.
An idempotent is called primitive if it is nonzero and cannot be written as the sum of two orthogonal nonzero idempotents.
An idempotent ''e'' is called a block or centrally primitive if it is nonzero and central and cannot be written as the sum of two orthogonal nonzero central idempotents. In this case the ideal ''eR'' is also sometimes called a block.
If the identity 1 of a ring ''R'' can be written as the sum
:1 = ''e''
1 + ... + ''e''
''n''
of orthogonal nonzero centrally primitive idempotents, then these idempotents are unique up to order and are called the blocks or the ring ''R''. In this case the ring ''R'' can be written as a direct sum
:''R'' = ''e''
1''R'' + ... + ''e''
''n''''R''
of indecomposable rings, which are sometimes also called the blocks of ''R''.
References
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External links
* {{springer, title=Peirce decomposition, id=p/p071970
Peirce decompositiono
http://www.tricki.org/
Algebras