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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**

- The
*even integers*2**Z**form a subring of**Z**.

More generally, if*n*is any integer the set of all multiples of*n*is a subring*n***Z**of**Z**.

The odd integers do not form a subring of**Z**. - The subsets {0, 2, 4} and {0, 3} are subrings of
**Z**_{6}. - 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. - The set {
*a*+*b*√5 |*a*,*b*∈**Z**} is a subring of the ring**R**. - The set {
*x*+*y*√5 |*x*,*y*∈**Q**} is also a subring of**R**. - The set of real matrices of the form forms a subring of the ring of
*all*2 × 2 real matrices.

An

**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**

- Examples 1) and 2) of subrings are also ideals, while examples 3), 4), 5) and 6) are not.
- 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*^{-1}*a*and so 1 is in too.

- 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. - 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.

- 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.

Here is a very important way of making ideals.

**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* > = {*r*_{1}*s*_{1} + *r*_{2}*s*_{2} + ... + *r*_{k}*s*_{k} ∈ *R* | *r*_{1} , *r*_{2} , ... ∈ *R*, *s*_{1} , *s*_{2} , ... ∈ *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**

- It is easy to see that the above does define an ideal.
- 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*> . - 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**

- The ideal 2
**Z**of**Z**is the*principal ideal*< 2 >. - Example 4 above (the polynomials in
**R**[*x*] with 0 constant term) is the principal ideal <*x*> . - 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. - The set of polynomials in
**R**[*x*,*y*] with zero constant coefficient is the ideal <*x*,*y*> and is not principal. - 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.

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