# Rings

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

• 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 is a ring, then we can define the set of polynomials (basically arithmetic expressions in the variable ) and think of any polynomial giving rise to a function defined by substituing in for in and computing using and as defined in .

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

such that

• gives the structure of an abelian group

• is associative and commutative with identity

• distributes over . I.e., for all .