semiprimelike

Semiprimelike is a term encountered in ring theory and related algebraic contexts. It is not a standard or universally agreed-upon notion, but rather a label attached to several loosely connected ideas that generalize or approximate the notion of a semiprime object. Because the term lacks a single accepted definition, its exact meaning can vary between authors and topics.

Two common pilot definitions appear in the literature. First, an element a in a ring R is called semiprimelike if the principal ideal Ra it generates is semiprime, meaning that whenever x^n lies in Ra for some n ≥ 1, then x itself lies in Ra. Equivalently, the radical of Ra equals Ra, so Ra is a semiprime ideal. Second, a ring R may be termed semiprimelike (or all elements be semiprimelike) if Ra is semiprime for every a ∈ R. Under this reading, every principal ideal has the semiprime property.

Example: In the ring of integers Z, the principal ideal (n) is semiprime precisely when n is square-free (no squared prime divides n). Thus, in Z, an integer n is semiprimelike in the sense of Definition A exactly when n is square-free (up to units).

Relationships and notes: Semiprime ideals and square-free generators connect semiprimelike concepts to radical theory and prime decompositions. The term remains nonstandard, and different sources may favor different emphases—such as focusing on elements, ideals, or the behavior under ring homomorphisms. When encountering semiprimelike, it is advisable to check the specific definition used in that context.

See also: semiprime, prime ideal, radical, ring theory, reduced ring.