Mathematicians are generally more familiar with the case of local commutative rings, so we’ll begin from there.

Definition. A commutative ring R is said to belocalif it has a unique maximal ideal.

Note that every non-zero commutative ring has at least one maximal ideal; indeed, we can take the collection ∑ of all ideals of *R* not containing 1; Zorn’s lemma provides a maximal element *I *in ∑, which does not contain 1 and is maximal.

Lemma. If R is a commutative local ring with maximal ideal M, then its complement R-M is the set of units of R.

**Proof**

If *x*∈*R* is a unit, it cannot be contained in the proper ideal *M*, so it must lie in *R*–*M*. Conversely, if *x*∈*R* is not a unit, the ideal *Rx* is properly contained in *R* so it is contained in a maximal ideal. But since there’s only one maximal ideal we have *Rx* ⊆ *M*, so *x*∈*M*, and we have *x*∈*R*–*M*. ♦

Proposition. A commutative ring R is local if and only if whenever x and y are non-units, so is x+y. [ Equivalently, if x+y is a unit, then either x or y is a unit. ]

**Proof**

Suppose *R* is local with maximal ideal *M*. The above lemma says *M* is precisely the set of non-units of *R*. Thus this set is closed under addition.

Conversely, let *M* be the set of non-units of *R* and assume (*x*, *y* ∈ *M* ⇒ *x*+*y* ∈ *M*). We claim that *M* is an ideal. First it is closed under addition by condition. Next if *x*∈*M*, *y*∈*R*, we claim *xy*∈*M*; indeed if not, *xy* is a unit so both *x* and *y* are units (contradiction!). So the set of non-units of *R* is an ideal. On the other hand, every *x*∈*R*–*M* is a unit so cannot be contained in a maximal ideal. This shows that *M* is the unique maximal ideal of *R*. ♦

**Examples**

1. Let *R* be the set of rational numbers of the form *a*/*b*, where *b* is not divisible by 7. Then *R* is local; indeed, *x*∈*R* is a unit if and only if *x*=*a*/*b*, where *a*, *b* are not divisible by 7. So if *x*,*y*∈*R* are non-units, then *x*=*a*/*b*, *y*=*c*/*d*, where *a*,* b* are divisible by 7 and *c*, *d* are not. Hence, so is *x*+*y* = (*ad*+*bc*)/(*bd*). Thus, the unique maximal ideal *M* of *R* is given by *a*/*b*, where *a* is divisible by 7 and *b* is not.

2. Let *R* be the set of rational functions in *x* of the form *f*(*x*)/*g*(*x*) where *f*(*x*), *g*(*x*) are polynomials and *g* has non-zero constant term. An element of *R* is non-unit iff it is of the form *f*(*x*)/*g*(*x*) where *f* has zero constant term and *g* does not. By the same reasoning as in example 1, this forms the unique maximal ideal of *R*.

3. More generally let *S* be an integral domain and *P* be any prime ideal of *S*. Take *K*, the field of fractions of *S*. Now define:

The set of non-units of *R* is thus One sees that this is closed under addition (we need *P* to be prime so that the product of elements of *S*–*P* remains in *S*–*P*). So *R* is local with maximal ideal *M*.

Example 3 can even be generalized to a commutative ring with zero-divisors. This is the essence of *localization*, which is another story for another day.

## Non-commutative Local Rings

We repeat the above definition for non-commutative *R*.

Definition. Let R be a ring, not necessarily commutative. We say that R islocalif, whenever x, y ∈ R are non-unit, so is x+y.

In other words, *R* is local if whenever *x*+*y* is a unit, either *x* or *y* is a unit. Since *x* is a non-unit iff –*x* is, we see that the definition is unchanged when we replace *x*+*y* with *x*–*y.** *Hence, we have the following:

- (non-unit) ± (non-unit) = (non-unit);
- (unit) ± (non-unit) = (unit). [ For if it were a non-unit, bringing the LHS non-unit term to the RHS contradicts the first property. ]

Lemma. Let R be a local ring. If x, y are non-units, so is xy.

[ Note: this is not as trivial as it seems! Indeed, without the local property, we can only conclude that *x* has a right inverse. ]

**Proof**

Assuming *xy* is a unit, we get:

*y*non-unit, 1 unit ⇒ 1+*y*unit;*xy*unit,*x*non-unit ⇒*xy*+*x*unit;*xy*+*x*=*x*(*y*+1) unit,*y*+1 unit ⇒*x*unit (contradiction). ♦

Thus, in a local ring we have:

- (unit) × (unit) = (unit). [ Obvious. ]
- (non-unit) × (unit) = (non-unit). [ If this were a unit, divide by the inverse of the unit on the LHS to get a contradiction. ]
- (unit) × (non-unit) = (non-unit). [ Same as above. ]
- (non-unit) × (non-unit) = (non-unit). [ Above lemma. ]

Next we have:

Proposition. If R is a non-zero local ring, the set of non-units is precisely the Jacobian radical J(R), the set of “bad elements”.

**Proof**

Let *N* be the set of non-units of *R*. By definition, *N* is closed under addition/subtraction. By the above observation, we see that if *x* or *y* is a non-unit, so is *xy*. Thus *N* is a two-sided ideal of *R*.

Obviously no element of *J*(*R*) is a unit (for *R* must have a maximal left ideal). Thus *J*(*R*)⊆*N*. Conversely, if *x* lies in *N*, and not in some maximal left ideal *M*, we get *Rx* + *M* = *R*. So *yx* + *z* = 1 for some *y* in *R*, *z* in *M*. But *yx* and *z* are both non-units which sum up to a unit, contradicting what we know about local rings. ♦

Corollary. In an artinian local ring R, every element is either unit or nilpotent. The radical is precisely the set of nilpotent elements.

**Proof**

In an artinian ring, the Jacobian *J* is nilpotent. In particular, every element of *J* is nilpotent. But the above proposition says *J* is precisely the set of non-units! ♦

Finally, we have:

Lemma. Let R be a local ring, and I ⊂ R be a proper two-sided ideal. Then R/I is also local.

**Proof**

Suppose (*x*+*I*), (*y*+*I*) ∈ *R*/*I* are such that (*x*+*y*)+*I* is a unit, so there is *z*∈*R* such that *z*(*x*+*y*) – 1 and (*x*+*y*)*z* – 1 are elements of *I*. An element of *I* is not a unit. Since *R* is local, *z*(*x*+*y*) and (*x*+*y*)*z* are units (e.g. if *z*(*x*+*y*) and *z*(*x*+*y*)-1 are both non-units, then so is 1, which is absurd). Hence, *x*+*y* is a unit, since in general if *ab* and *ba* are both units, then *a* and *b* are both left- and right-invertible and units as well. Again since *R* is local, we know that *x* or *y* is a unit, and so *x*+*I* or *y*+*I* is a unit. ♦