The concept of prime ideal in a commutative ring with is one of several natural generalisations of the concept of prime integer number.

An ideal is called *prime* if for any the following holds:

implies that at least one of these elements, and is in

E.g. the principal ideal is prime if and only if is prime.

A maximal ideal is prime.

Indeed, let Assume that We need to show that then If this were not the case then and are two non-zero elements in the ring such that But this is not possible, as is a field. Thus as claimed.

Analysing this proof, one can easily see that

If is prime then has no zero-divisors, i.e. it is an integral domain.

Further important property of prime ideals is that they are radical, i.e.

where the radical of the ideal ideal is Indeed, implies that either or is in and we derive by applying this reduction.

Yet another interesting observation is that is *multiplicatively closed*.

**Rings of fractions**

A subset is called multiplicatively closed (or multiplicative) if and for any

Given a multiplicatively closed set one defines

a relation on as follows:

It is not hard to show that is an equivalence relation.

To simplify the notation, write its equivalence class with representative as We define

is the ring of fractions of w.r.t. with addition and multiplication given by the rules

It is easy to check that this is well-defined. We also have

so that is a ring homomorphism.

Note that need not be injective, i.e. need not hold. Indeed, if is a zero-divisor such that for then and so

The most well-known example is the case being an integral domain, and Then is a field, called the field of fractions of

**Examples:**

- is the field of fractions of
- for the ring of polynomials over a field the field is the field of rational functions over
- Let be non-nilpotent. Then isa multiplicative set. Moreover, then (it is not completely trivial to prove this, though). Intuitively, we make the variable behave like the inverse of as in this ring.

Now let us look at the case for a nonzero prime ideal. In this case is denoted by and called the localisation of at

** Example:** Let be the ring of polynomials over a field and Then is prime, and is equal to

The ring has unique maximal ideal

It suffices to show that is invertible in iff Indeed, if then On the other hand, if then there exists such that Thus and so (if it was, would be in as is an ideal).

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