Commutative rings

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

  1. (R,+) is an abelian group
  2. multiplication is associative, i.e. (a\cdot b)\cdot c=a\cdot (b\cdot c) for all a,b,c\in R
  3. multiplication is commutative, i.e. a\cdot b=b\cdot a for all a,b\in R
  4. multiplication is distributive over addition, i.e. a\cdot (b+c)=a\cdot b+a\cdot c for all a,b,c\in R

We can derive more useful rules from the above, e.g. that 0\in R (remember, R is an abelian group, and they all have the neutral element, 0) satisfies a\cdot 0=0.) This follows by computing a\cdot 0=a(b-b)=ab-ab=0 using distributivity.

Examples

  1. Integers: \mathbb{Z} and integers \mod m: \mathbb{Z}_m\cong \mathbb{Z}/m \mathbb{Z}
  2. Fields: e.g. \mathbb{R}, \mathbb{C}, \mathbb{Q}, \mathbb{Z}_p for a prime p\geq 2
  3. (Univariate) polynomials over a field \mathbb{F}: \mathbb{F}[t]=\{\sum_{i\geq 0} a_i t^i\mid a_i\in\mathbb{F}, a_k=0\text{ for all }k\geq d_a\}.
  4. Formal power series over a field \mathbb{F}: \mathbb{F}[[t]]=\{\sum_{i\geq 0} a_i t^i\mid a_i\in\mathbb{F} \}.
  5. Zero ring: take any abelian group A and define the trivial multiplication on it, i.e. a\cdot b=0 for all a,b\in A.
  6. Ring R of functions f: X\to \mathbb{R}, with addition and multiplication defined as (f+g)(x)=f(x)+g(x), (f\cdot g)(x)=f(x)\cdot g(x), for any f,g\in R.
  7. Functions f: \mathbb{R}\to \mathbb{R} 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 \mathbb{Z}, this is the case. Moreover, there need not be any multiplicative identity element: e.g. consider 2\mathbb{Z}, 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 identity is a commutative ring R that has a multiplicative identity element, denoted 1, so that a\dot 1=a for all a\in A.

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

0\neq a\in R is called a zero divisor if there exists 0\neq b\in R such that ab=0.

In a zero ring every non-0 is a zero divisor. Can you find a zero divisor in \mathbb{Z}_4?

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 1\neq 0 is an integral domain if it has no zero divisors.

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

A subring S of a commutative ring R is an abelian subgroup of (R,+) that is also closed under multiplication.

For instance, 2 \mathbb{Z} is a subring of \mathbb{Z}, the polynomials \mathbb{F}[t] are a subring of \mathbb{F}[[t]].
A very important class of subrings are ideals:

An ideal S of a commutative ring R is a subring of R that in addition satisfies the following: rS\subseteq S for any r\in R (here, as usual here, multiplication rS means \{rs\mid s\in S\}.

For instance, 2 \mathbb{Z} is an ideal of \mathbb{Z}, but \mathbb{F}[t] is not an ideal in \mathbb{F}[[t]].
R itself and \{0\} are ideals in R, for any commutative ring R. They are called improper ideals, and all the other ideals are called proper.

A commutative ring R with 1 is a field if and only if all its ideals are improper.

Indeed, if I is an ideal in R and 0\neq r\in I then, as R is a field, there exists r^{-1}\in R such that rr^{-1}=1. As I is an ideal, r^{-1}r=1\in I, implying I=R. On the other hand, if 0\neq r\in R does not have a multiplicative inverse, then Rr is a proper ideal in R.

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 I\subset R in a commutative ring R the set
S\subseteq I is called a generating set if a=\sum_{s\in S} r_s s, with r_s \in R and only finitely many of them non-0 for any a\in I. In other words, we can write I=\sum_{s in S} Rs.

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

an ideal I\subset R in a commutative ring R is called principal if I=Rs for an s\in I.

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