Directions. Although you may refer to books for definitions and standard theorems, searching for solutions to these written exercises is not permitted. You may not seek help from others.
Bear in mind that rings are always assumed to have a multiplicative identity, and a homomorphism of rings is always assumed to carry the multiplicative identity of its domain to that of its target. Recall that if is a ring, the term -algebra indicates, by definition, a pair where is a ring and is a ring homomorphism.
Let be a field, and let and be finite-dimensional vector spaces over . Recall that the ring of endomorphisms of an -vector space is an -algebra.
Explain why for and there is a unique for which
Show that the map given by is -bilinear.
Prove that the bilinear map in the previous part provides an isomorphism
Let be a field, an -dimensional vector space over , the vector space with the convention , and the direct sum Endow with the structure of a non-commutative -algebra as follows:
Define canonical bilinear maps
Use the bilinear maps of the previous item with standard facts about direct sums to define multiplication
With , , and as in the previous exercise, do the following:
Prove that if is -dimensional over , then is isomorphic to the polynomial ring .
State and prove a universal (initial) mapping property for the tensor algebra .
Let be a commutative ring and ideals in . Prove that
Let be a field, the unique -algebra homomorphism for which and the unique -algebra homomorphism for which .
Let be the -vector space having basis which may be canonically identified with the subspace . Let be the unique -algebra homomorphism for which and be the unique -algebra homomorphism for which .
Show that is universal-initial among triples where is a centered -algebra, and are -algebra homomorphisms satisfying for all .
Show that is universal-initial among triples where is a centered -algebra, and are any -algebra homomorphisms.