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

 

[blue_raver22]

I can provide a response to this question from a mathematical perspective. In ring theory, unity refers to the existence of a multiplicative identity element within a ring. An integral domain is a type of ring where every nonzero element has a multiplicative inverse.

When unity is taken in integral domains, it means that the ring has a multiplicative identity element, and all nonzero elements have a multiplicative inverse. This allows for certain properties and theorems to hold, such as the cancellation law and the uniqueness of the multiplicative inverse. However, when unity is not taken in integral domains, the ring may not have these properties, and certain theorems may not hold.

In terms of results becoming more general in the second case, it really depends on the specific problem and theorems being considered. In some cases, not assuming unity may lead to more general results, while in others, it may limit the scope of the problem.

As for why there is not one standard way adopted worldwide by all authors, it is because there may be different ways of approaching a problem and different perspectives on how to define and use certain concepts. Additionally, different authors may have different preferences or may be working within different contexts, leading to variations in the use of terminology and notation. Ultimately, as long as the mathematical truth is accurately expressed and understood, the specific notation or approach may not be as important.