abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, 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 (just notation, not necessarily th ...

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

\mathbb_S(T)=\.

In ring theory, if ''A'' is an additive subgroup of a

ring (The) Ring(s) 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 Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

''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 identi ...

(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 \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

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 unswapped ...

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 ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

, 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 (order), lattice ''x'' ≤ ''y'' and a monoid ''x''•''y'' which admits operations ''x''\''z'' and ''z''/''y'', loosely analogous to division or i ...

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 ...

''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 contin ...

''A'' is

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...

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

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

''H''.

Notes

References

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