Document Type
Article
Department
Mathematics (HMC)
Publication Date
1977
Abstract
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 [1] 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).
Rights Information
© 1977 European Mathematical Society
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Recommended Citation
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.
Comments
Previously linked to as: http://ccdl.libraries.claremont.edu/u?/irw,448.
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This article is also available at http://purl.pt/2832.