Prove that a commutative ring is nonsingular if and only if it is semiprime. If R is the ring of

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Prove that a commutative ring is nonsingular if and only if...

Key Concepts

Maximal Quotient Rings

The maximal quotient ring, sometimes referred to as the classical ring of quotients, is an enlargement of a given ring obtained by ‘inverting’ a suitable set of elements. This construction is useful for studying rings in a more controlled environment where certain troublesome elements are removed, thereby elucidating properties such as regularity and the absence of minimal ideals in an extended context.

Semiprime Rings

A semiprime ring is one that does not possess any nonzero nilpotent ideals, meaning that the intersection of all its prime ideals is zero. This condition guarantees that the ring does not contain ‘hidden’ nilpotency, which otherwise might lead to complications in the structure theory of rings. For commutative rings, semiprimeness often coincides with desirable properties such as being nonsingular.

Von Neumann Regular Rings

A ring is called von Neumann regular if every element a in the ring has an element b such that a = aba. This property implies that the ring behaves very much like a field in terms of solving equations, albeit not necessarily possessing multiplicative inverses for every nonzero element. It is a significant concept in ring theory because it often leads to systems with ‘nice’ decomposition properties and a well-understood ideal structure.

Commutative Rings

A commutative ring is an algebraic structure where addition and multiplication are defined, with multiplication being commutative. This foundational concept in ring theory ensures that many properties, such as ideals and quotient rings, behave symmetrically, thus simplifying discussions concerning singularity, regularity, and various quotient constructions.

Nonsingular Rings

A ring is said to be nonsingular if it has no nonzero elements whose annihilators are ‘large’ in a module-theoretic sense. In other words, elements have ‘small’ annihilators, preventing pathological behavior when forming quotient modules. This definition is essential in understanding how the ring avoids degenerate cases or ‘singular’ behavior in module actions.

Minimal Ideals

Minimal ideals are the nonzero ideals that do not contain any proper subideals besides the zero ideal and themselves. The presence or absence of minimal ideals gives insight into the building blocks of the ring. A ring lacking minimal ideals often indicates a kind of ‘spread-out’ structure where the fine-grained decomposition into simple components is not possible, which can be a consequence of the ring being von Neumann regular.

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