What is the meaning of $\mathbb{Z}[a], a\in \mathbb{Z}$

Question

What is the meaning of $\mathbb{Z}[a], a\in \mathbb{Z}$ (for instance $\mathbb{Z}[2])$. I know about integer rings $\mathbb{Z}_p$ and also about $\mathbb{Z}[x]$, but I've never seen the former notation.

Answer 1

Usually if $S$ is a ring, $R\subseteq S$ a subring and $A\subseteq S$ a subset then

$$R[A]=\bigcap\{R'\subseteq S\ |\ R' \text{ is a subring such that } R\subseteq R', A\subseteq R'\}$$ $$R[a]=R[\{a\}]\text{ for }a\in S$$

In other words $R[A]$ is the smallest subring of $S$ that contains both $R$ and $A$.

Note that if $A\subseteq R$ then you can easily check that $R[A]=R$.

Some notable examples:

1. $\mathbb{R}[i]=\mathbb{C}$
2. $\mathbb{Z}[a]=\mathbb{Z}$ for any $a\in\mathbb{Z}$
3. $\mathbb{Z}[1/2]=\{p/2^n\ |\ p,n\in\mathbb{Z}\}$
4. $\mathbb{Q}[\pi]\simeq\mathbb{Q}[X]$ with the ring of polynomials on the right side (note "$\simeq$" instead of "$=$"). That's because $\pi$ is transcendental.

Generally if $R\subseteq S$ and $a\in S$ then you have a ring epimorphism

$$R[X]\to R[a]$$ $$X\mapsto a$$

with the ring of polynomials on the left side. Therefore you can look at $R[a]$ as a quotient ring

$$R[a]\simeq R[X]/I$$

for some ideal $I$. Thus you effictively get rid of $S$ in the sense that the definition of $R[a]$ is independent on $S$. It only depends on additive/multiplicative properties of $a$ itself.

This can be generalized to non-singleton extensions: $R[A]\simeq R[X_i]/I$ with the ring of polynomials in multiple variables on the right side.