Introduction to Ring Theory with Examples in Abstract Algebra (including Subrings)

For Ring Theory in Abstract Algebra, we define a ring R to be a nonempty set with two binary operations, addition + and multiplication *, defined on it (implying closure). The ring R is an Abelian group under addition +. Furthermore, under multiplication *, two associative properties are satisfied as well as the distributive property of multiplication over addition. To make sure we include matrix ring examples, we do NOT require that multiplication be commutative. When multiplication is commutative, the ring R is called a commutative ring. We also do not require R to have a multiplicative identity. When R has a multiplicative identity, we call it a unity and write it as "1". The integers ℤ forms a commutative ring with unity under ordinary addition and multiplication. The even integers 2ℤ forms a commutative ring without unity under ordinary addition and multiplication. The integers mod n, ℤn = {0,1,2,3,...,n-1}, forms a commutative ring with unity under addition and multiplication mod n. Beware! {0,2,4} is a subring of ℤ6 without "1" in it, but it still has a unity. The unity of {0,2,4} is "4" under multiplication mod 6.

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[kristianwichmann]

A ring with a single element is a group under multiplication.