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Fred Rohrer
Fred Rohrer
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There exists a non-noetherian ring $R$ such that every non-zero $R$-module has an associated prime.

This in proven in Example 2.3 in P. J. Cahen, Ascending chain conditions and associated primes, Commutative ring theory (Fès, 1992), 41-46, Lecture Notes in Pure and Appl. Math., 153, Dekker, New York, 1994.

The point is that over an arbitrary commutative ring, any non-zero module has a weakly associated prime. Thus, the question is related to asking for rings over which assassins and weak assassins coincide. Clearly, noetherian rings have this property, but the aforementioned example shows that there are also other such rings. I do not know any more classes of such rings, nor about any other (published) work on these questions. See also this question.

There exists a non-noetherian ring $R$ such that every non-zero $R$-module has an associated prime.

This in proven in Example 2.3 in P. J. Cahen, Ascending chain conditions and associated primes, Commutative ring theory (Fès, 1992), 41-46, Lecture Notes in Pure and Appl. Math., 153, Dekker, New York, 1994.

The point is that over an arbitrary commutative ring, any non-zero module has a weakly associated prime. Thus, the question is related to asking for rings over which assassins and weak assassins coincide. Clearly, noetherian rings have this property, but the aforementioned example shows that there are also other such rings. I do not know any more classes of such rings, nor about any (published) work on these questions. See also this question.

There exists a non-noetherian ring $R$ such that every non-zero $R$-module has an associated prime.

This in proven in Example 2.3 in P. J. Cahen, Ascending chain conditions and associated primes, Commutative ring theory (Fès, 1992), 41-46, Lecture Notes in Pure and Appl. Math., 153, Dekker, New York, 1994.

The point is that over an arbitrary commutative ring, any non-zero module has a weakly associated prime. Thus, the question is related to asking for rings over which assassins and weak assassins coincide. Clearly, noetherian rings have this property, but the aforementioned example shows that there are also other such rings. I do not know any more classes of such rings, nor about any other (published) work on these questions. See also this question.

Source Link
Fred Rohrer
Fred Rohrer
  • 6.7k
  • 1
  • 27
  • 44

There exists a non-noetherian ring $R$ such that every non-zero $R$-module has an associated prime.

This in proven in Example 2.3 in P. J. Cahen, Ascending chain conditions and associated primes, Commutative ring theory (Fès, 1992), 41-46, Lecture Notes in Pure and Appl. Math., 153, Dekker, New York, 1994.

The point is that over an arbitrary commutative ring, any non-zero module has a weakly associated prime. Thus, the question is related to asking for rings over which assassins and weak assassins coincide. Clearly, noetherian rings have this property, but the aforementioned example shows that there are also other such rings. I do not know any more classes of such rings, nor about any (published) work on these questions. See also this question.