What are a cyclic $k$-algebra and a monoid $k$-algebra?

I am trying assignments of algebraic geometry from an online course and these definitions were not mentioned in theory.

So, can you please help me with understanding them?

1. Let $k$ be a field. Let $B= k[x]$ be a cyclic $k$-algebra.

What does it mean for a $k$-algebra to be cyclic? A $k$-algebra is a pair of a ring and ring homomorphism.

2. Let $ B = k [ {\mathbb{N}}^2]$ be the monoid algebra over $k$ of the additive monoid $\mathbb{N}^2$.

So, can you please help me with these definitions?

Solution 1:

1. Cyclic means single generator. In this case, that generator is $x$. (It is very common to take the homomorphism $k\to k[x]$ to be the "standard" one taking $1\mapsto 1$ and extending $k$-linearly and not discuss this at all, if you're worried about that part of things.)

2. It's just like a group algebra, but with monoids instead. In particular, given a ring $R$ and a monoid $\mathcal{M}$, the monoid algebra $R[\mathcal{M}]$ is the free $R$-module on the elements of $\mathcal{M}$ with multiplication defined by applying the monoid operation: that is, if $a,b\in\mathcal{M}$ with associated basis elements $e_a$ and $e_b$ respectively, then $e_a\cdot e_b=e_{a\cdot b}$. You can check that in your specific case, $k[\Bbb N^2]\cong k[x,y]$ with $x$ corresponding to $(1,0)$ and $y$ corresponding to $(0,1)$.