The set of nilpotent elements in a commutative ring is an ideal, and it is called the nilradical. Another equivalent description is that it is the intersection of the prime ideals. It could be the zero ideal, as in the case of the integers.
A commutative regular ring is characterized by its semiprime ideals. At the end we demonstrate that the intersection of all fuzzy semiprime ideals of a commutative ring which contain a given fuzzy ideal λ is precisely the nil radical √λ of λ. This establishes the vital connection between ...
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quasi-nil-ideals i. e. Ω/N is semi-simple. In §2 some elemlntary prorerties of the radical of a general ring Ω are inve- stigated and the following theorems are proved: 1°The radical of a ring Ω is the intersection of all the two-sided ideals M of Ω such that Ω/M is...
We define s-prime ideals of M and show that N(M) is equal to the intersection of the s-prime ideals of M. If R is a ring, the nil radical of R considered as a Γ-ring with Γ = R is equal to the upper nil radical of R. We also give a sufficient condition for the ...
A ring R is semiprime (semiprimitive) if and only the intersection of the prime (primitive) ideals is zero. Then, R is a subdirect product of prime (primitive) rings 26.6 and 26.13. The (McCoy) prime radical of a ring is defined to be the intersection of the prime ideals, and is ...
Necessary and sufficient conditions are provided for the nilpotence of the intersection of all τ-closed prime ideals of a ring R with τ-Krull dimension, where τ denotes a hereditary torsion theory on the category Mod-R of right R-modules. It is shown that these conditions hold, in ...
In this paper a generalization of the notion of m -system set of rings to modules is given. Then for a submodule N of M , we define := { m 蔚 M every m -system containing m meets N }. It is shown that is the intersection of all prime submodules of M containing N . We ...
semiprime moduleclassical m-system setlower nilradicalBaeru2013McCoy radicalstrongly nilpotentleft Goldie ringLet M be a left R-module. A proper submodule P of M is called classical prime if for all ideals ${mathcal{A}}, {mathcal{B}} subseteq R$ and for all submodules N u2286 M, ${...
Then for a submodule N of M, we define $nsqrt[p]{N}n$nsqrt[p]{N}n:= { m 蔚 M: every m-system containing m meets N}. It is shown that $nsqrt[p]{N}n$nsqrt[p]{N}n is the intersection of all prime submodules of M containing N. We define rad R (M) = $nsqrt[p]...