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.

 

[blue_raver22]

I can provide some insight into the different definitions of integral domain from Herstein and other sources. It is important to note that in mathematics, different authors may use slightly different definitions for the same concept. In this case, both definitions are valid and are used in different contexts.

Herstein's definition of an integral domain does not assume the existence of a multiplicative unit element, which means that the ring may not have an element that acts as a multiplicative identity. This definition is commonly used in abstract algebra, where the focus is on studying the properties of rings without assuming any additional structure.

On the other hand, the more common definition of an integral domain does assume the existence of a multiplicative unit element. This definition is often used in other branches of mathematics, such as number theory, where the concept of a unit element is more relevant.

Both definitions have their own advantages and are used in different contexts. It is important to understand that the concept of an integral domain remains the same regardless of the definition used. The key property of an integral domain is that it does not have any non-zero elements whose product is zero, which is captured in both definitions.

In conclusion, it is not a matter of one definition being more common than the other, but rather a matter of different contexts and purposes for which the concept is being used. As a scientist, it is important to understand and be familiar with different definitions and their applications in order to effectively use them in our research and studies.