Idealizer

In abstract algebra, the '''idealizer''' of a subsemigroup ''T'' of a semigroup ''S'' is the largest subsemigroup of ''S'' in which ''T'' is an ideal.{{sfn|Mikhalev|Pilz|2002|loc=p.30}} Such an idealizer is given by

:<math>\mathbb{I}_S(T)=\{s\in S \mid sT\subseteq T \text{ and } Ts\subseteq T\}.</math>

In ring theory, if ''A'' is an additive subgroup of a ring ''R'', then <math>\mathbb{I}_R(A)</math> (defined in the multiplicative semigroup of ''R'') is the largest subring of ''R'' in which ''A'' is a two-sided ideal.{{sfn|Goodearl|1976|loc=p.121}}{{sfn|Levy|Robson|2011|loc=p.7}}

In Lie algebra, if ''L'' is a Lie ring (or Lie algebra) with Lie product [''x'',''y''], and ''S'' is an additive subgroup of ''L'', then the set

:<math>\{r\in L\mid [r,S]\subseteq S\}</math>

is classically called the '''normalizer''' 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''] ⊆ ''S'', because anticommutativity of the Lie product causes [''s'',''r''] = −[''r'',''s''] ∈ ''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, :<math>\mathbb{I}_R(T)=\{r\in R \mid rT\subseteq T \}</math> if ''T'' is a right ideal, or :<math>\mathbb{I}_R(L)=\{r\in R \mid Lr\subseteq L \}</math> if ''L'' is a left ideal.

In commutative algebra, 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 :<math>(A:B):=\{r\in R \mid Br\subseteq A\}</math>. In terms of this conductor notation, an additive subgroup ''B'' of ''R'' has idealizer :<math>\mathbb{I}_R(B)=(B:B)</math>.

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

;Examples The multiplier algebra ''M''(''A'') of a C*-algebra ''A'' is isomorphic to the idealizer of ''π''(''A'') where ''π'' is any faithful nondegenerate representation of ''A'' on a Hilbert space ''H''.

==Notes== {{Reflist}}

==References== *{{citation |last=Goodearl|first=K. R. |title=Ring theory: Nonsingular rings and modules |series=Pure and Applied Mathematics, No. 33 |publisher=Marcel Dekker Inc. |place=New York |year=1976 |pages=viii+206 |mr=0429962}} *{{citation |last1=Levy|first1=Lawrence S. |last2=Robson |first2=J. Chris|title=Hereditary Noetherian prime rings and idealizers |series=Mathematical Surveys and Monographs |volume=174 |publisher=American Mathematical Society |place=Providence, RI |year=2011 |pages=iv+228 |isbn=978-0-8218-5350-4|mr=2790801}} *{{citation |title=The concise handbook of algebra |editor1-last=Mikhalev |editor1-first=Alexander V. |editor2-last=Pilz |editor2-first=Günter F. |publisher=Kluwer Academic Publishers |place=Dordrecht |year=2002 |pages=xvi+618 |isbn=0-7923-7072-4 |mr=1966155}}

Category:Abstract algebra Category:Group theory Category:Ring theory