Polynomial algebra and polynomial ring

Question

What is the difference between polynomial algebra and polynomial ring? because sometimes I read polynomial algebra and it looks like a polynomial ring $K[x,y,..]$ in many variables.

Thanks

Answer 1

A ring is a set, $S$ with two operations on it, $+ \colon S \times S \to S$ and $\ast \colon S\times S \to S$ that behave in a nice way.

An algebra is a vector space, $V$ over a field, $F$ of scalars with three operations $+ \colon V \times V \to V, \ast \colon F \times V \to V$ and $\circ \colon V \times V \to V$ where the operations behave in a nice way.

Functionally there is no difference, as every scalar is a polynomial.

Answer 2

Assuming everything is commutative, a polynomial algebra is the same thing as a polynomial ring. A polynomial ring just happens to be (the canonical example of) an algebra (for which you should look up the definition; essentially it's a ring with coefficients in another ring), so it's correct to use either term. It's just a matter of taste.

Answer 3

A ring is defined on two operations which are called addition and multiplication. The definition of a $K$-algebra is based on a commutative unitary ring which acts as scalars on a ring $A$. So not every $K$-algebra is a ring but every ring is a $K$-algebra by using $K$ as the integers $\mathbb{Z}$.