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 substituing 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

• $(+):R\times R\to R$

• $(\cdot ):R\times R\to R$

• $(-):R\to R$

• $0\in R$

• $1\in R$

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 $x,y,z$.