Some Sufficient Conditions for the Jacobson Radical of a Commutative Ring with Identity to Contain a Prime Ideal
Throughout, the word "ring" will abbreviate the phrase "commutative ring with identity element 1" unless the contrary is stated explicitly. An ideal I of a ring R is called pseudoprime if ab = 0 implies a or b is in I. This term was introduced by C. Kohls and L. Gillman who observed that if I contains a prime ideal, then I is pseudoprime, but, in general, the converse need not hold. In [9 p. 233], M. Larsen, W. Lewis, and R. Shores ask if whenever the Jacobson radical J(R) of an arithmetical ring is pseudoprime, it follows that J(R) contains a prime ideal? In Section 2, I answer this question affirmatively. Indeed, if R is arithmetical and J(R) is pseudoprime, then the set N(R) of nilpotent elements of R is a prime ideal (Corollary 9). Along the way, necessary and sufficient conditions for J(R) to contain a prime ideal are obtained. In Section 3, I show that a class of rings introduced by A. Bouvier  are characterized by the property that every minimal prime ideal of R is contained in J(R). The remainder of the section is devoted to rings with pseudoprime Jacobson radical that satisfy a variety of chain conditions. In particular, it is shown that if R is a Noetherian multiplication ring with pseudoptime Jacobson radical J(R), then J(R) contains a unique minima] prime ideal (Theorem 20), but there is a NoetheIian semiprime ring R such that J(R) is pseudoprime and fails to contain a prime ideal (Example 21).
© 1977 European Mathematical Society
Henriksen, Melvin. "Some sufficient conditions for the Jacobson radical of a commutative ring with identity to contain a prime ideal." Portugaliae Mathematica 36.3-4 (1977): 257-269.
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