Rings

A ring is like a field, but where elements may not be invertible. Basically, it's a structure where we can

• add

• multiply

• subtract

but not necessarily divide. If you know what polynomials are already, you can think of it as the minimal necessary structure for polynomials to make sense. That is, if $R$ is a ring, then we can define the set of polynomials $R[x]$ (basically arithmetic expressions in the variable $x$) and think of any polynomial $f \in R[x]$ giving rise to a function $R \to R$ defined by substituting in for $x$ in $f$ and computing using $+$ and $\cdot$ as defined in $R$.

So, in full, a ring $R$ is a set equipped with

• $(+) \colon R \times R \to R$

• $(\cdot) \colon R \times R \to R$

• $(-) \colon R \to R$

• $0 \in R$

• $1 \in R$

such that

• $(+, 0, -)$ gives the structure of an abelian group

• $(\cdot)$ is associative and commutative with identity $1$

• $+$ distributes over $\cdot$. I.e., $x \cdot (y + z) = x\cdot y + x \cdot z$ for all $x, y, z$.