Subrings and ideals

These are the concepts which play the same role as subgroups and normal subgroups in group theory.

Definition

A subring S of a ring R is a subset of R which is a ring under the same operations as R.

Equivalently: The criterion for a subring
A non-empty subset S of R is a subring if a, b ∈ S ⇒ a - b, ab ∈ S.

So S is closed under subtraction and multiplication.

Exercise: Prove that these two definitions are equivalent.

Remark

Using the above criterion makes it easy to check that something is a ring by showing that it is a subring of something else since one does not need to check associativity or distributivity.

Examples

1. The even integers 2Z form a subring of Z.
   More generally, if n is any integer the set of all multiples of n is a subring nZ of Z.
   The odd integers do not form a subring of Z.

2. The subsets {0, 2, 4} and {0, 3} are subrings of Z₆.

3. The set {a + bi ∈ C | a, b ∈ Z} forms a subring of C.
   This is called the ring of Gaussian integers (sometimes written Z[i]) and is important in Number Theory.

4. The set {a + b√5 | a, b ∈ Z} is a subring of the ring R.

5. The set {x + y√5 | x, y ∈ Q} is also a subring of R.

6. The set of real matrices of the form forms a subring of the ring of all 2 × 2 real matrices.
   

Definition

A subring I of R is a left ideal if a ∈ I, r ∈ R ⇒ ra ∈ I.

So I is closed under subtraction and also under multiplication on the left by elements of the "big ring".

A right ideal is defined similarly.

A two-sided ideal (or just an ideal) is both a left and right ideal.
That is, a, b ∈ I, r ∈ R ⇒ a - b, ar, ra ∈ I.

Remark

These subsets are related to the ideal numbers that Eduard Kummer (1810 to 1893) defined to "restore the uniqueness of factorisation" in the rings used for proving cases of Fermat's last theorem.

Examples

1. Examples 1) and 2) of subrings are also ideals, while examples 3), 4), 5) and 6) are not.

2. In any ring R the subsets {0} and R are both two-sided ideals. If R is a field these are the only ideals.
   Proof
   Note that if the identity 1 is in an ideal then the ideal is the whole ring. But if a field element a ≠ 0 is in an ideal, so is a^-1a and so 1 is in too.
   

3. The set of real matrices of the form forms a left ideal of the ring of all 2 × 2 real matrices while those of the form form a right ideal of this ring.
   This ring does not have any proper non-trivial two-sided ideals.

4. The set of all polynomials over any ring with 0 "constant" coefficient form an ideal.
   Proof
   Such a polynomial is of the form xq(x) for some polynomial q(x) and it is easy to verify the ideal condition for these.
   

5. The set of all polynomials in Z[x] whose coefficients are all even is an ideal. So is the set of those with even constant coefficient.
   

Definitions

Let R be a commutative ring with identity. Let S be a subset of R. The ideal generated by S is the subset < S > = {r₁s₁ + r₂s₂ + ... + r_ks_k ∈ R | r₁ , r₂ , ... ∈ R, s₁ , s₂ , ... ∈ S, k ∈ N}.

In particular, if S has a single element s this is called the principal ideal generated by s.

That is, < s > = {rs | r ∈ R}.

Remarks

1. It is easy to see that the above does define an ideal.

2. If the ring is not commutative then the above defines a left ideal. It is easy to modify the definition to get a right ideal or a two-sided ideal. If the ring does not have an identity then in general S will not be a subset of < S > .

3. In general, the "thing" generated by a subset is the smallest "thing" containing the subset. So you can talk about the subgroup of a group generated by a subset or the subring of a ring generated by a subset, ...
   

Examples

1. The ideal 2Z of Z is the principal ideal < 2 >.

2. Example 4 above (the polynomials in R[x] with 0 constant term) is the principal ideal < x > .

3. The set of all polynomials in Z[x] whose coefficients are all even is the principal ideal < 2 >.
   The set of all polynomials with even constant coefficient is the ideal < 2, x > and is not principal.

4. The set of polynomials in R[x, y] with zero constant coefficient is the ideal < x, y > and is not principal.

5. For any commutative ring with identity, the trivial ideal {0} is the principal ideal < 0 > and the whole ring is the principal ideal < 1 >.
   

We will see later that in the rings Z and R[x] every ideal is principal.