## Definition and Basics of Ideals

**Point of Post: **In this post we define left, right, and two-sided ideals.

*Motivation*

As was alluded to when we discussed subrings, subrings themselves aren’t the important thing when discussing rings, in the same way that it’s not subgroups but normal subgroups that play a pivotal role in group theory. In particular, what we’d eventually like to do is to define the quotient objects for rings–the ‘quotient rings’. But, as normal we can’t quotient out by any old subset, or even any old subring, of a ring. It turns out, much the same as the case for groups that what defines these special subrings is that they are the kernels of ring homomorphisms. But, this is all in the future. What we shall show in this post is that kernels of ring homomorphisms satisfy a certain ‘absorption’ property and then define these analogs of normal subgroups as being subrings that satisfy this property

*Ideals*

We begin by noting that kernels of ring homomorphisms satisfy a certain absorption principle which is much stronger than being multiplicatively closed. Namely:

**Theorem: ***Let and be rings and a ring homomorphism. Then, for any and any it’s true that .*

**Proof: **We merely note that and and so as claimed.

We abstactify this property and call a subring of a ring an *ideal *if whenever and . We generally denote ideals by lower case Fraktur letters such as . More generally we call a subring of a ring a *left ideal *if whenever and , *right ideals *are defined similarly. When convenient we shall call an ideal a *two-sided ideal.*

Let . We define (i.e. the set of all finite combinations of elements of with coefficients in ). We define similarly as the set as well as (this last one evidently being equal to ). If (i.e. if is a singleton) we denote , and as , and respectively.

For any ring there are always two ideals, namely and . We call these the *trivial ideals of *. An ideal is *proper *if , a proper left and right ideal are defined similarly.

There are some fundamental facts about ideals, in particular:

**Theorem: ***Let and be rings and , then is an ideal of .*

**Theorem: ***Let be a ring and , then , , and is a left, right, and two-sided ideal respectively.*

**Proof: **We merely note that and thus is a left ideal as claimed. The other two cases are done similarly.

**Theorem: ***Let be a unital ring and a left or right ideal. Then, if and only if contains a unit.*

**Proof: **Evidently if then . Conversely, if then for any one has that or depending on whether is a left or right ideal, either way and so the conclusion follows.

**Corollary: ***Let be a ring and an ideal. Then, if and only if contains a unit.*

**Theorem: ***Let be a unital ring. Then, is a division alebra if and only if the only left ideals of are the trivial ones.*

**Proof: **

**Lemma: ***Let be a monoid where every element has a left inverse . Then, every element of has a two-sided inverse and –in other words is a group.*

**Proof: **Let be arbitrary. Since it suffices to prove that . To do this merely note that

So is a two-sided inverse of . Since was arbitrary the conclusion follows.

Suppose first that has no non-trivial left ideals. Since is a multiplicative monoid we see from the lemma it suffices to show that every element of is left invertible. So, let be arbitrary. We know from the above theorems that is a left ideal which is non-zero (since ) and thus by assumption . But, in particular this means that and so there exists some for which , and thus is left invertible. Since was arbitrary the conclusion follows.

Conversely, suppose that is a division ring and let be a non-zero left ideal. Then, by assumption there exists some but by definition and thus contains a unit, so by the above theorem we have that .

**Corollary: ***Let be a commutative unital ring. Then, is a field if and only if all ideals are trivial.*

**References:**

1. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. *Basic Abstract Algebra*. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

R^\times is a division ring this term is not true

Comment by mohamad | June 21, 2011 |

Mohamad,

I assume you meant the “Conversely, suppose that is a division ring..” in the last proof? If so, it was indeed a typo, it clearly should have been “Conversely, suppose that is a division ring…”. It’s fixed now! Thank you very much for pointing that out!

Best,

Alex

Comment by Alex Youcis | June 21, 2011 |

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