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.