4ring

4ring is a term that can appear in different contexts, but in mathematics it is often used to refer to a ring with four elements. There is no single canonical object universally called “4ring,” but the phrase is commonly associated with the two nonisomorphic rings of order four that appear in finite ring theory.

In the standard algebraic classification, the rings of order four are, up to isomorphism, the ring of integers modulo 4, denoted Z4, and the direct product ring Z2 × Z2. Z4 has elements {0, 1, 2, 3}, with addition and multiplication performed modulo 4. It is a commutative ring with identity, but not a field, and it has characteristic 4. It contains a zero divisor (for example, 2 · 2 = 0), and its unit group consists of {1, 3}. Z4 is a principal ideal ring, and its ideals are {0}, <2>, and the whole ring.

Z2 × Z2 consists of four elements arranged as pairs: {(0,0), (1,0), (0,1), (1,1)} with componentwise addition and multiplication. This ring is also commutative with identity, but it is not a domain and has zero divisors. It is a Boolean ring, since every element is idempotent (x^2 = x for all x). Its unit group contains only the element (1,1). The ideal structure is straightforward, with four ideals corresponding to the combinations of ideals in each component.

Beyond mathematics, the term 4ring can appear as a brand or product name in various domains, but its exact meaning is context dependent. See also finite rings, ring theory, Z4, and Z2 × Z2.