Principal ideal

In the mathematical field of ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x of P, which is to say the set of all elements less than or equal to x in P.

The remainder of this article addresses the ring-theoretic concept.

While this definition for two-sided principal ideal may seem to contrast with the others, it is necessary to ensure that the ring remains closed under addition.

If R is a commutative ring, then the above three notions are all the same. In that case, it is common to write the ideal generated by a as ⟨a⟩.

Not all ideals are principal. For example, consider the commutative ring C[x,y] of all polynomials in two variables x and y, with complex coefficients. The ideal ⟨x,y⟩ generated by x and y, which consists of all the polynomials in C[x,y] that have zero for the constant term, is not principal. To see this, suppose that p were a generator for ⟨x,y⟩; then x and y would both be divisible by p, which is impossible unless p is a nonzero constant. But zero is the only constant in ⟨x,y⟩, so we have a contradiction.

In the ring ${\displaystyle Z[{\sqrt {-3}}]=\{a+b{\sqrt {-3}}:a,b\in Z\}}$, numbers in which a + b is even form a non-principal ideal. This ideal forms a regular hexagonal lattice in the complex plane. Consider (a,b) = (2,0) and (1,1). These numbers are elements of this ideal with the same norm (2), but because the only units in the ring are 1 and -1, they are not associates.

...
Wikipedia