Rings, units, and zero divisors

A ring has an abelian group under addition: sums associate and commute, there is an additive zero, and every element has an additive inverse. Multiplication associates, has an identity called one, and distributes over addition on both sides. Our commutative convention also requires multiplication to commute. The integers, rational numbers, real numbers, complex numbers, and polynomials over any of these are examples. Matrix rings satisfy the more general ring axioms but usually fail commutativity, so they fall outside this module's standing convention.

These axioms explain several identities rather than assuming them separately. Distributing a product over zero plus zero gives the same product twice, and additive cancellation then forces a product with zero to be zero. Similarly, multiplying by an additive negative produces the negative of the product. A ring can therefore support subtraction perfectly well while lacking division. The zero ring has one equal to zero; we allow it as a ring but exclude it from fields and integral domains.

A unit is an element with a multiplicative inverse inside the ring. The integer units are one and negative one: their integer products can equal one. Every nonzero rational number is a unit in the rationals. Thus being a unit describes an element together with its ambient ring. A nonzero integer such as two becomes a unit after passing to a larger ring, but it was not already invertible in the smaller one.

A nonzero element is a zero divisor if it annihilates some other nonzero element. In the residue ring modulo six, the classes of two and three are zero divisors because their product is the zero class. A unit cannot be a nonzero zero divisor: multiply an alleged zero product by its inverse. However, a nonunit need not be a zero divisor. Two in the integers is a basic counterexample, and prevents us from confusing failure of division with failure of cancellation.

Suppose a times b equals a times c. Subtracting the two products gives a times the difference equal to zero. If a is nonzero and not a zero divisor, the difference must be zero, so cancellation works. In the integers every nonzero factor is cancellable even though most factors are not units. Cancellation is therefore weaker than the ability to divide within the ring. Both deserve their own justification in a proof.

Modulo six, two times one and two times four are both two, yet one and four are different classes. Canceling two here would manufacture a false conclusion. When working in an unfamiliar algebra, first ask whether it has zero divisors and whether the specific coefficient is a unit. These questions become essential in polynomial equations, quotient constructions, and linear systems with modular coefficients, where procedures copied from real arithmetic can otherwise fail silently.