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In a noetherian ring, why is it true that there are only a finite number of minimal prime ideals of some ideal? (And is it proven somewhere in the Atiyah-mcdonald book?)
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A finite prime ideal in a Noetherian ring is a subset of the ring that is closed under addition, subtraction and multiplication, and is also an ideal. It is a prime ideal if it cannot be expressed as the intersection of two larger ideals, and is finite if it contains a finite number of elements.
Finite prime ideals play an important role in the study of Noetherian rings, which are commutative rings with certain desirable properties. They allow for the decomposition of a Noetherian ring into smaller, more manageable components and are key in proving important theorems such as the primary decomposition theorem.
In a Noetherian ring, an element is considered irreducible if it cannot be written as the product of two non-invertible elements. Finite prime ideals correspond to irreducible elements in the ring, with each finite prime ideal being generated by a single irreducible element.
No, a Noetherian ring cannot have an infinite number of finite prime ideals. This is because a Noetherian ring must satisfy the ascending chain condition, meaning that any sequence of ideals in the ring must eventually stabilize and become constant. This implies that there can only be a finite number of distinct finite prime ideals in a Noetherian ring.
Finite prime ideals in Noetherian rings have important connections to algebraic geometry, number theory, and representation theory. They also have applications in coding theory and cryptography. Additionally, the study of finite prime ideals has led to the development of new mathematical techniques and tools that have been used to solve various problems in these areas.