Introduction to Ring Theory (2)

Subrings

Just like groups have subgroups, we have:

Definition. A subset S of a ring R is a subring if it satisfies the following:

$1\in S$ ;

$r,s \in S \implies r-s\in S$ ;

$r,s\in S \implies rs\in S$ .

The first two conditions imply that S is a subgroup of (R, +). Together with the third condition, this means S inherits a ring structure from (R, +, ×) via the same addition and product operations. Clearly, if T is a subring of S which is in turn a subring of R, then T is a subring of R.

Examples

The ring of integers Z has no proper subring: since a subring must contain 1, it must contain all integers. Same goes for Z/n for positive integer n.
We have the sequence of subrings $\mathbf{Z} \subset \mathbf{Q} \subset \mathbf{R} \subset \mathbf{C}$ .
In the ring of 2 × 2 real matrices, the set U of upper-triangular matrices $\begin{pmatrix} * & * \\ 0 & *\end{pmatrix}$ forms a subring, and the set D of diagonal matrices forms a subring of U. [ Note that D is isomorphic to R × R, under component-wise addition and multiplication. We haven’t defined isomorphism, but you know what it means. 🙂 ]
In the ring of quaternions, the subset of elements of the form a+bi is a subring which is isomorphic to C. Thus, we have a commutative subring of a non-commutative ring. Specifically, we have a division ring with a subring which forms a field.
In the ring R × R, the subset {(r, r) : r in R} is a subring which is isomorphic to R.
In the ring of polynomials R[x], take the subset R[x²] comprising of polynomials whose terms are all even powers of x. This is a subring.

In the ring R × R, the subset R × {0} is not a subring, because it doesn’t contain the unity (1, 1). The reader may object that this ought to be a bona-fide subring since it’s isomorphic to R which is a ring with unity, but we wish to maintain consistency with the definition. In other words, what we have here is a “subrng” which is isomorphic to a ring, but still not a subring.

Incidentally, some books do not require a ring to possess a unity 1, in which case R × {0} would become a subring of R × R. Both rings would then have unities but the two unities are different elements. This can lead to much grief if we weren’t careful, which is why we would rather not deal with that case.

Generated Subrings

Basic property: the intersection of subrings is also a subring. Explicitly, if {S_i} is a collection of subrings of R, then ∩_iS_i is also a subring of R. The proof is standard:

Since 1 is in each S_i, it also lies in all ∩_iS_i.
If x, y are in ∩_iS_i, then they are in S_ifor all i; since each S_iis a subring, x–y lies in S_i, and so x–y lies in ∩_iS_itoo.
For multiplication, same as above, but replace x-y with xy. ♦

If X is an arbitrary subset of R, we let <X> be the intersection of all subrings of R which contain X. As in the case of groups, <X> is a subring of R containing X, and conversely any subring S of R containing X must contain <X>. We call this the subring of R generated by X.

Consider the field C of complex numbers. What’s the subring S generated by the empty set? Well S must contain 1, so since S is closed under addition it must also contain all integers. On the other hand, the set of all integers itself is already a subring. So S = Z.
Take the field C of complex numbers again. What’s the subring S generated by a single element 1/2? As before S must contain all integers. But now S must also contain all powers of 1/2 since it’s closed under multiplication. Therefore S contains all elements of the form a/2^m, where a is an integer and m is a non-negative integer. On the other hand, the set of all such elements forms a subring. So $S= \{a/2^m : a \in\mathbf{Z}, m\in\mathbf{Z}, m\ge 0\}$ .
Again take C. What’s the subring S generated by a single element i (i.e. √-1)? Now S contains all integers and also contains i. So it must contain all numbers of the form a+bi, where a, b are integers. Numbers of this form are called Gaussian integers. It’s clear that the set of Gaussian integers forms a subring, so S = set of Gaussian integers, denoted by Z[i].
Once again, take C. What’s the subring S generated by a single element π? Since S contains Z and π, and S is closed under addition and multiplication, S must contain all real numbers of the form $a_0 + a_1 \pi + \ldots + a_n \pi^n$ , $a_i\in\mathbf{R}$ . On the other hand, the set of such numbers clearly forms a subring, so S = set of all numbers of the form $\sum_{i=0}^n a_i \pi^n$ . Furthermore, it is known that π is transcendental, so different expressions give distinct numbers. To be specific, there is an isomorphism Z[x] → S, mapping x to π.
Now take R[x]. What’s the subring S generated by x²? Again S contains Z and the element x² so it must contain all polynomials in x² with integer coefficients. This means S = Z[x²]. On the other hand, we can also ask for the subring generated by R and x². This just means the subring generated by $\mathbf{R}\cup\{x^2\}$ . Now it’s clear that the resulting subring is R[x²].

Exercise. Consider the ring of all 2 × 2 real matrices.

Describe the subring generated by $\begin{pmatrix} 2 & 1\\ 1 & 2\end{pmatrix}$ .
Describe the subring generated by $\begin{pmatrix} 1 & 1\\{0} & 1\end{pmatrix}$ .
Describe the subring generated by both $\begin{pmatrix} 2 & 1\\ 1 & 2\end{pmatrix}$ and $\begin{pmatrix} 1 & 1\\{0} & 1\end{pmatrix}$ .