One operation only often does not capture the whole mathematical picture. Two, or even more, operations are often used, and you have in fact seen examples of such objects in previous courses, such as fields, polynomials, power series, square matrices. What unites these examples together is that they have an abelian group lurking in the background, and a multiplication operation that need not be too nice (with the exception of fields.) In fact, matrix multiplication isn’t ever commutative; we postpond investigating the latter to another course.

A

commutative ringis a set with two binary operations, addition and multiplication, denoted by and respectively, (for multiplication the dot is often omitted all together) that satisfy the following axioms:

- is an abelian group
- multiplication is associative, i.e. for all
- multiplication is commutative, i.e. for all
- multiplication is distributive over addition, i.e. for all

We can derive more useful rules from the above, e.g. that (remember, is an abelian group, and they all have the neutral element, 0) satisfies ) This follows by computing using distributivity.

**Examples**

- Integers: and integers :
- Fields: e.g. for a prime
- (Univariate) polynomials over a field :
- Formal power series over a field :
- Zero ring: take any abelian group and define the trivial multiplication on it, i.e. for all
- Ring of functions with addition and multiplication defined as for any
- Functions that possess a property preserved under addition and multiplication form a ring (e.g. ring of continuos functions, ring of differentiable functions).

Note that we talk about polynomials and power series not as functions, but as formal (that’s why “formal” in formal power series!) expressions we know how to add and multiply.

So we see that commutative rings are like fields, but with less demands placed on multiplication: there need not be the multiplicative inverse to any non-0 element, for instance, in this is the case. Moreover, there need not be any multiplicative identity element: e.g. consider the set of all even integers with the usual addition and multiplication; it does not have a multiplicative identity. So we separate the cases when there is a multiplicative identity into a separate case:

A

commutative ring with identityis a commutative ring that has a multiplicative identity element, denoted so that for all

Another difference is that is can happen in a ring that for two non-0 elements Such elements have a special name:

is called a

zero divisorif there exists such that

In a zero ring every non-0 is a zero divisor. Can you find a zero divisor in

The fields are certainly the nicest kind of rings. The second best do not have zero divisors and have a “real” 1:

A commutative ring with identity is an

integral domainif it has no zero divisors.

**Subrings and ideals**

After seeing the definition of a subgroup, the following should not be a surprise:

A

subringof a commutative ring is an abelian subgroup of that is also closed under multiplication.

For instance, is a subring of the polynomials are a subring of

A very important class of subrings are ideals:

An

idealof a commutative ring is a subring of that in addition satisfies the following: for any (here, as usual here, multiplication means

For instance, is an ideal of but is not an ideal in

itself and are ideals in for any commutative ring They are called *improper* ideals, and all the other ideals are called *proper*.

A commutative ring with is a field if and only if all its ideals are improper.

Indeed, if is an ideal in and then, as is a field, there exists such that As is an ideal, implying On the other hand, if does not have a multiplicative inverse, then is a proper ideal in

Again, as in the case of abelian groups, we can talk about generating sets of subrings and ideals. (Certainly, we shall use both operations to express elements in terms of generators.) In the case of ideals, the natural definition is as follows:

In an ideal in a commutative ring the set

is called a generating set if with and only finitely many of them non-0 for any In other words, we can write

The analog of a cyclic subgroup, an ideal with a generating set of size 1, is called *principal*:

an ideal in a commutative ring is called

principalif for an

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