# Integral Domain

1. Aug 14, 2014

### gianeshwar

Dear Friends,
Please tell me the differences created in ring theory problems when
1.Unity is taken in integral domains.
2. Unity is not taken in integral domains.
Do results become more general in the second case.
Why one standard way not adopted worldwide by all authors because mathematical truth must be expessed in only one standard way .

Last edited: Aug 14, 2014
2. Aug 14, 2014

### mathman

3. Aug 14, 2014

### mathwonk

p.199, hungerford, shows every ring embeds in a ring with identity element. so there seems to be no greater generality or interest.

4. Aug 14, 2014

### jbunniii

Some familiar facts that we learned at our mother's knee become false if you remove the 1.

For one thing, maximal ideals are no longer automatically prime ideals. For example, $R = 2\mathbb{Z}$ is now an integral domain, and $I = 4\mathbb{Z}$ is a maximal ideal which is not prime, since $4 = 2\cdot 2$ and $4\in I$ but $2 \not\in I$.

Worse yet, $R$ need not even have any maximal ideals. See, e.g.

http://sierra.nmsu.edu/morandi/notes/NoMaxIdeals.pdf