By David S. Dummit, Richard M. Foote
Commonly acclaimed algebra textual content. This ebook is designed to provide the reader perception into the ability and sweetness that accrues from a wealthy interaction among diversified parts of arithmetic. The e-book conscientiously develops the idea of alternative algebraic constructions, starting from simple definitions to a few in-depth effects, utilizing various examples and routines to help the reader's figuring out. during this method, readers achieve an appreciation for the way mathematical constructions and their interaction bring about robust effects and insights in a few diversified settings.
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Additional resources for Abstract Algebra
Let f : A → B be a homomorphism of O-algebras, let b be an ideal of A , and let e be a primitive idempotent of A which does not belong to Ker(f ) . Then e ∈ b if and only if f (e) ∈ f (b) . Proof. We have f (e) ∈ f (b) if and only if e ∈ b + Ker(f ) . By Rosenberg’s lemma, this is equivalent to e ∈ b because e ∈ / Ker(f ) . If f : A → B is a homomorphism of O-algebras, the image of a primitive idempotent of A is in general not a primitive idempotent of B . The easiest example occurs when A = O and f is the natural map making B into an O-algebra: the image of the primitive idempotent 1O is 1B , which decomposes according to the points of B and their multiplicities (defined below).
Recall that an A-module P is called projective if it is a direct summand of a free A-module, or equivalently, if for every surjective homomorphism f : M → N , any homomorphism g : P → N lifts to a homomorphism g : P → M such that f g = g . In fact it is sufficient to assume this when g = id , that is, to require that any surjective homomorphism f : M → P splits. Recall also that an A-module I is called injective if for every injective homomorphism f : M → N , any homomorphism g : M → I extends to a homomorphism g : N → I such that g f = g .
Thus f is O-linear and satisfies f (ab) = f (a)f (b) for all a, b ∈ A . If a homomorphism f : A → B satisfies f (1A ) = 1B , then f is called unitary. In the general case f (1A ) is an idempotent of B and the image of f is contained in the subalgebra f (1A )Bf (1A ) of B . For example if e is an idempotent of A , the inclusion of eAe into A is a homomorphism. It is in fact precisely in order to be able to consider these inclusions that one does not require homomorphisms to be unitary. So another way of visualizing a homomorphism f : A → B is to view it as a unitary homomorphism f : A → eBe , for some idempotent e of B , followed by the inclusion eBe → B .
Abstract Algebra by David S. Dummit, Richard M. Foote