Polynomial Rings (Pt. III)
Point of Post: This post is a continuation of this one.
Just as the case for the product of rings and matrix rings there is a certain ‘functorial’ property for matrix rings. In particular given two commutative unital rings and a morphism there is a natural induced map given by . Indeed:
Theorem: Let and be commutative unital rings and be a ring morphism. Then, the map given by is a ring morphism which is a unital morphism if is. Furthermore, and . Moreover, carries monomorphisms to monomorphisms and epimorphisms to epimorphisms, and thus isomorphisms to isomorphisms.
Proof: The fact that is a morphsims which is unital if is (it’s just a calculation) as well as the fact that is obvious. To prove that we note that for any one has
The fact that carries monomorphisms to monomorphisms, epimorphisms to epimorphisms, and isomorphisms to isomorphisms is obvious.
Remark: In other words, for people who like to phrase it in this way, the map by taking and is an endofunctor.
From this we get the two useful corollaries:
Corollary: Let and be commutative unital rings, then implies
Corollary: Let be a commutative unital ring and an ideal of . Then, is an ideal of and .
Polynomial Rings in Finitely Many Indeterminates
Let be a commutative integral, we define the polynomial ring over in the indeterminates , denoted , inductively by . In other words, is the set of all polynomials in the indeterminate with coefficients in . So, the elements of are of the form
where denotes if and . In this context is known as the multidegree of the monomial and the degree of the monomial is defined by . For an arbitrary polynomial we define the degree of to be the maximum of the degrees of all its monomial terms.
The only other thing we note about polynomial rings in finitely many indeterminates, besides the definition and degree, is that by applying the two previous results concerning one can prove that is an integral domain if and only if is, and that the units of are precisely the polynomials.
1. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.
2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. Basic Abstract Algebra. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.