The awkward middle child of algebra

Abstract algebra is basically the study of groups, rings, and fields. There are more concepts, but these are the big three.

Groups have the least structure and fields have the most structure. Rings are somewhere in the middle.

Groups have just one operation, which is thought of as multiplication by default. If the group operation is commutative, it’s thought of as addition. But in either case, there’s only one operation.

Fields have two operations, addition and multiplication, and the operations have all the basic properties you expect.

Rings also have two operations, addition and multiplication, but they lack some of the familiar properties of fields.

The most awkward thing about ring theory is that there is no universal agreement on the definition of a ring. The disagreement is over whether the definition should require the existence of a multiplicative identity, the analog of the number 1. Most authors include this as part of the definition, but a substantial portion do not.

There are two ways of dealing with this mess. The most common is to require rings to have an identity, but continually remind readers of this convention. And so you’ll see things like “Let R be a ring (with identity).”

The other less common solution is to give custody of the term “ring” to those who require an identity and use the term “rng” for the thing that’s just like a ring except it doesn’t necessarily have an identity.

Another awkward, or at least curious, feature of ring theory is that you almost never speak of subrings. Subgroups are important in group theory, and subfields are important in field theory [1]. But you almost never hear the word “subring.”

The ring theory counterpart to subgroups in group theory is the ideal. The name comes from Earnst Kummer thinking of these objects as ideal numbers, numbers added to an algebraic system to fill in some deficiency.

Given a ring R, a subset I of R is an ideal if it is closed under addition and under multiplication by any element of R. Note the asymmetry between addition and multiplication. You can add two elements of I and stay in I. And you can multiply an element in I, not just by another element of I, but by any element in R, and stay in I.

A quick example will help. Let R be the integers and let I be all multiples of 7. The sum of two multiples of 7 is another multiple of 7, And if you multiply a multiple of 7 by any integer you get another multiple of 7. So multiples of 7 form an ideal in the integers.

The disagreement over whether to require identities and the lack of interest in subrings are related. If you don’t require a ring to have an identity element, then you don’t require subrings to have an identity element, and so every ideal is a subring. But if you do require rings to have an identity, the only ideal of R that is a subring is R itself.

A final awkward thing about ring theory is that there are so many kinds of rings: integral domains, principle ideal domains, Noetherian rings, Artinian rings, etc. It seems that “ring” itself is too generic to be useful except as a pedagogical device to organize what more specific kinds of rings have in common.

In this sense ring theory is analogous to “absolute geometry,” the study of the common features of Euclidean and Non-Euclidean geometry. In practice one cares about more specific geometries, but there’s some value in cataloging theorems that various geometries share.

Related posts

• Visualizing kinds of rings
• Finite fields
• Finite simple groups

[1] The concept of a subfield is more common than the term “subfield.” In field theory, it’s more common to say that K is an extension field of F than to say that F is a subfield of K, though the two statements are equivalent.