Completely Prime Ideal Rings and Their Extensions
نویسنده
چکیده
Let R be a ring and let I 6= R be an ideal of R. Then I is said to be a completely prime ideal of R if R/I is a domain and is said to be completely semiprime if R/I is a reduced ring. In this paper, we introduce a new class of rings known as completely prime ideal rings. We say that a ring R is a completely prime ideal ring (CPI-ring) if every prime ideal of R is completely prime. We say that a ring R is a near completely prime ideal ring (NCPI-ring) if every minimal prime ideal of R is completely prime. We say that a ring R is an almost completely prime ideal ring (ACPI-ring) if every associated prime ideal of R (R viewed as a right module over itself) is completely prime. Let now R be a Noetherian ring which is also an algebra over Q (Q is the field of rational numbers) and δ a derivation of R. Then we prove the following: (1) R is a near completely prime ideal ring if and only if R[x; δ] is a near completely prime ideal ring. (2) R is an almost completely prime ideal ring if and only if R[x; δ] is an almost completely prime ideal ring.
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