abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...

, the idealizer of a subsemigroup ''T'' of a

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'' ...

''S'' is the largest subsemigroup of ''S'' in which ''T'' is an ideal. Such an idealizer is given by :

\mathbb_S(T)=\.

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 rep ...

, if ''A'' is an additive subgroup of a ring ''R'', then

\mathbb_R(A)

(defined in the multiplicative semigroup of ''R'') is the largest subring of ''R'' in which ''A'' is a two-sided ideal. In

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

, if ''L'' is a

Lie ring In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

(or

) with Lie product 'x'',''y'' and ''S'' is an additive subgroup of ''L'', then the set :

\

is classically called the

normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...

of ''S'', however it is apparent that this set is actually the Lie ring equivalent of the idealizer. It is not necessary to specify that 'S'',''r''nbsp;⊆ ''S'', because

anticommutativity In mathematics, anticommutativity is a specific property of some non- commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswappe ...

of the Lie product causes 's'',''r''nbsp;= − 'r'',''s''nbsp;∈ ''S''. The Lie "normalizer" of ''S'' is the largest subring of ''L'' in which ''S'' is a Lie ideal.

Comments

Often, when right or left ideals are the additive subgroups of ''R'' of interest, the idealizer is defined more simply by taking advantage of the fact that multiplication by ring elements is already absorbed on one side. Explicitly, :

\mathbb_R(T)=\

if ''T'' is a right ideal, or :

\mathbb_R(L)=\

if ''L'' is a left ideal. 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 ...

, the idealizer is related to a more general construction. Given a commutative ring ''R'', and given two subsets ''A'' and ''B'' of a right ''R''-module ''M'', the conductor or transporter is given by :

(A:B):=\

. In terms of this conductor notation, an additive subgroup ''B'' of ''R'' has idealizer :

\mathbb_R(B)=(B:B)

. When ''A'' and ''B'' are ideals of ''R'', the conductor is part of the structure of the

residuated lattice In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice ''x'' ≤ ''y'' and a monoid ''x''•''y'' which admits operations ''x''\''z'' and ''z''/''y'', loosely analogous to division or implication, when ...

of ideals of ''R''. ;Examples The

multiplier algebra In mathematics, the multiplier algebra, denoted by ''M''(''A''), of a C*-algebra ''A'' is a unital C*-algebra that is the largest unital C*-algebra that contains ''A'' as an ideal in a "non-degenerate" way. It is the noncommutative generalization of ...

''M''(''A'') of a

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...

''A'' is

isomorphic 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 ...

to the idealizer of ''π''(''A'') where ''π'' is any faithful nondegenerate representation of ''A'' on a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

''H''.

Notes

References

* * * Abstract algebra Group theory Ring theory {{Abstract-algebra-stub