## Polynomial Rings (Pt. III)

**Point of Post: **This post is a continuation of this one.

*Isomorphism Considerations*

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 .*

**Proof: **Consider the epimorphism induced from the canonical epimorphism . Note then that the kernel of this map is evidently and so the result follows immediately from the first isomorphism theorem.

*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.

**References:**

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.

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