1, the principal ideal ring is a ring that makes every ideal generated by a single element in mathematics. If every ideal of the whole ring D is a principal ideal, it is called a principal ideal ring; If the principal ideal ring is also an integral ring, it is called the principal ideal integral ring, which is often abbreviated as PID.

2. Definition of equivalence: Let A be an integral ring, then the following conditions are equivalent: A is an integral ring of the principal ideal, and every prime ideal of A is the existence norm of the principal ideal A. ..

3. Theorem of related properties: The ideals generated by irreducible elements in the principal ideal ring are maximal ideals. The principal ideal integral ring is the only factorized integral ring. In the principal ideal ring d, let d be the greatest common factor of a and b, then < a, b>=<d>. Let d be the greatest common factor of a and b in the main ideal ring d, then? U, v∈D makes: au+bv = D. Unique decomposability Let R be the principal ideal ring, then any non-zero element a∈R can be uniquely decomposed.

4. Example analysis: Integer rings are principal ideal rings, more generally speaking, Euclidean rings are always principal ideal rings. Polynomial rings over fields are principal ideal rings. Gaussian integer rings are principal ideal rings. Eisenstein integer ring is a principal ideal ring, where ω is any cubic unit root except 1. Rings that are not principal ideal rings can prove that ideals cannot be generated by a single element.