Abstract algebra | Ring theory | Group theory
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. Such an idealizer is given by In ring theory, if A is an additive subgroup of a ring R, then (defined in the multiplicative semigroup of R) is the largest subring of R in which A is a two-sided ideal. 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 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. (Wikipedia).
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We prove that every ideal of a quotient of a principal ideal domain is also principal. Notice that the new space may not be an integral domain, so it is sometimes called a principal ring. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal Website: http:
From playlist Abstract Algebra
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After introducing the notion of a principal ideal domain (pid), we give some examples, and prove some simple results. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal Website: http://www.michael-penn.net Randolph College Math: http://www.randolphcolle
From playlist Abstract Algebra
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From playlist Abstract Algebra
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