Ring Theory Problems: Unity vs. Non-Unity

[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 .

 

[mathman]

http://en.wikipedia.org/wiki/Integral_domain

See the introduction.

 

[mathwonk]

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

 

[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