Definition of integral domain from Herstein

[mikepol]

Hi,

I ran into conflicting definitions of integral domain. Herstein defines a ring where existence of unity for multiplication is NOT assumed. His definition of integral domain is:

"A commutative ring R is an integral domain if ab=0 in R implies a=0 or b=0"

I looked in 3 other books and on the Internet, and everywhere either integral domain is defined to contain a multiplicative unit element, or definition of a ring assumes such an element. In either case, integral domain seems to always contain a unit element.

Could someone please explain to me why are there two different definitions and which one is more common?

Thank you.

 

[mansurbhimsen]

My understanding is that a unity element is a prerequisite for defining an integral domain in general. However as Herstein shows in the proof for the theorem that every finite integral domain is a field, its easy to show the inherent existence of an unity element in an integral domain when its finite. The pigeonhole principle and associated logic used in that proof relies on the integral domain being finite.I doubt its possible to show the existence of unity in an integral domain if its infinite without assuming it already. I am waiting for a better insight myself . Hope this suffices till then.

 

[mansurbhimsen]

a bit of additional info.its a cut and paste job from wikipedia...but it seems to answer some questions nevertheless

"In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0.[1] That is, it is a ring which has no left or right zero divisors. (Sometimes such a ring is said to "have the zero-product property.") Some authors require the ring to have 1 ≠ 0 to be a domain,[2] or equivalently to be nontrivial[3]. In other words, a domain is a nontrivial ring without left or right zero divisors. A commutative domain with 1 ≠ 0 is called an integral domain.[4]

A finite domain is automatically a finite field by Wedderburn's little theorem."

 

[mathwonk]

I looked in a dozen or so books (Lang, Rotman, Dummit and Foote, Van der Waerden, Reid, Matsumura, Atiyah - Macdonald, Zariski - Samuel, Eisenbud, Herstein, Brauer, A.A.Albert, Birkhoff-Maclane) and found several different conventions.

In general the reason for giving a definition in a certain way is that the author finds it convenient for what he intends to do in his own book. Or maybe he just imitates what he was taught. Or maybe he feels that his definition captures the most interesting examples out there.

Anyway, a nice theorem in the book of Jacobson (and maybe also Hungerford) is that every ring without an identity can be embedded isomorphically as an ideal inside a ring with identity. Moreover every ring without zero divisors can be embedded isomorphically inside a ring without zero divisors which has an identity element.

I guess I forgot to check whether commutativity is assumed. Anyway the result suggests that you never need to exclude the identity element since every ring without one is actually an ideal in a ring with one. So considering rings with identity along with their ideals covers the whole territory.

Jacobson cleverly calls rings without identity "rngs".

you might check me on this in hungerford or jacobson.