Let k be a commutative ring. (i) Prove that k[x, y] is a...

Let $k$ be a commutative ring. (i) Prove that $k[x, y]$ is a free commutative $k$-algebra with basis $\{x, y\}$. (ii) Use Proposition $5.2$ to prove that $k[x] \otimes_{k} k[y]$ is a free $k$ algebra with basis $\{x, y\}$. (iii) Use Proposition $5.4$ to prove that $k[x] \otimes_{k} k[y] \cong k[x, y]$ as $k$-algebras.

Let $k$ be a commutative ring.
(i) Prove that $k[x, y]$ is a free commutative $k$-algebra with basis $\{x, y\}$.
(ii) Use Proposition $5.2$ to prove that $k[x] \otimes_{k} k[y]$ is a free $k$ algebra with basis $\{x, y\}$.
(iii) Use Proposition $5.4$ to prove that $k[x] \otimes_{k} k[y] \cong k[x, y]$ as $k$-algebras.

Key Concepts

Commutative Ring

A commutative ring is an algebraic structure in which addition and multiplication are defined and behave similarly to the integers; in particular, addition forms an abelian group and multiplication is associative, has an identity, and is distributive over addition. The commutativity of multiplication is a central feature, which allows many constructions in algebra, such as polynomial rings and tensor products, to have well-behaved properties.

Polynomial Ring

A polynomial ring in one or more variables over a commutative ring is a ring formed by polynomials whose coefficients come from that ring. It serves as a fundamental example of a free commutative algebra and is characterized by its universal property: any function from the set of variables to a commutative algebra over the base ring uniquely extends to a homomorphism of algebras.

Free Commutative Algebra

A free commutative algebra over a ring is an algebra that is characterized by having a basis in the sense that any function from a set (the basis) to any commutative algebra uniquely extends to an algebra homomorphism. The polynomial ring in several variables is a prime example, as its construction does not impose any relations among the variables other than those required by commutativity and the ring axioms.

Tensor Product of Algebras

The tensor product of algebras over a commutative ring provides a way to combine two algebras into a new one in a bilinear fashion. It plays a crucial role in algebra by allowing constructions that merge different algebraic structures, and under suitable conditions, the tensor product of free algebras exhibits properties that mirror those of polynomial rings in multiple variables.

Universal Mapping Property

The universal mapping property is a defining characteristic of free objects in a category, such as polynomial rings or free algebras. It states that for any function from the generating set to any other object in the category, there is a unique morphism that extends this function. This property is fundamental in establishing various isomorphisms and constructions, as it provides a canonical way to relate different algebraic structures.

Transcript

Please give Ace some feedback