An example of a ring without identity that does not contain any maximal ideal.

Question

I'm trying to find an example of a ring without identity that does not contain any maximal ideal.

Help me some hints.

Since you don't insist that there is an identity, just take an abelian group and define all products to be 0. This is a ring without identity and any subgroup is an ideal. So an abelian group that contains no maximal proper subgroup would be an example. — KCd, Dec 25 '13 at 05:46
Continuing with KCd's remark, consider the abelian group $\mathbb{Q}$. — Prism, Dec 25 '13 at 08:35
I would try to reproduce the properties of a field. What happens if you have ax=c always having a solution? If this doesn't imply identity then we have only trivial ideals. — Jacob Wakem, Mar 16 '14 at 22:08

score 7 · Accepted Answer · answered Dec 25 '13 at 09:43

7

Take an abelian group without maximal subgroups, like $(\Bbb Q,+)$, and the zero multiplication, that is, $xy=0$ for any $x,y$.

answered Dec 25 '13 at 09:43

I've posted this trivial example in order to prevent future possible trivial questions on this theme. – Dec 25 '13 at 12:26

score 3 · Answer 2 · answered Dec 25 '13 at 10:16

3

answered Dec 25 '13 at 10:16

Truong

2 Answers2