Cryptography: Zn is more than a commutative ring, but not quite an integral domain. Why?

Because Zn contains a multiplicative identity element. Commutative rings are not required to possess multiplicative identities.

Why is Zn not an integral domain?
Even though Zn possesses a multiplicative identity, it does NOT satisfy the other condition of integral domains which says that if a × b = 0 then either a or b must be zero.
Consider modulo 8 arithmetic. We have 2 × 4 = 0, which is a clear violation of the second rule for integral domains.

Cryptography: Groups, Abelian Group, Ring, Commutative Ring, Integral Domain, Fields

Cryptography: Rings: Integral Domain

Rings: Integral Domain

Integral domain, is a commutative ring that obeys the following axioms.

•(M5) Multiplicative identity: There is an element 1 in R such that a1 = 1a = a for all a in R.

•(M6) No zero divisors: If a, b in R and ab = 0, then either a = 0 or b = 0.

Let S be the set of integers, positive, negative, and 0, under the usual operations of addition and multiplication. S is an integral domain.

•ADDITIONAL PROPERTY 1: The set R must include an identity element for the multiplicative operation. That is, it should be possible to symbolically designate an element of the set R as ’1’ so that for every element a of the set we can say a1 = 1a = a

•ADDITIONAL PROPERTY 2: Let 0 denote the identity element for the addition operation. If a multiplication of any two elements a and b of R results in 0, that is if ab = 0 then either a or b must be 0.

Rings: Integral Domain - Example 

•The set of all integers under the operations of arithmetic addition and multiplication.

•The set of all real numbers under the operations of arithmetic addition and multiplication.