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An approach for the calculation of a Gröbner basis into several smaller computations following a path in the Gröbner fan of the ideal generated by the system of inequalities.

In a local ring R, there is only one maximal ideal m. Hence, R has only one quotient ring R/m which is a field. This field is called the residue field.

The kernel of a ring homomorphism f:R-->S is the set of all elements of R which are mapped to zero. It is the kernel of f as a homomorphism of additive groups. It is an ideal ...

Given a number field K, a Galois extension field L, and prime ideals p of K and P of L unramified over p, there exists a unique element sigma=((L/K),P) of the Galois group ...

A popular acronym for "principal ideal domain." In engineering circles, the acronym PID refers to the "proportional-integral-derivative method" algorithm for controlling ...

Let P be a prime ideal in D_m not containing m. Then (Phi(P))=P^(sumtsigma_t^(-1)), where the sum is over all 1<=t<m which are relatively prime to m. Here D_m is the ring of ...

A von Neumann regular ring is a ring R such that for all a in R, there exists a b in R satisfying a=aba (Jacobson 1989, p. 196). More formally, a ring R is regular in the ...

Baer's criterion, also known as Baer's test, states that a module M over a unit ring R is injective iff every module homomorphism from an ideal of R to M can be extended to a ...

Every Boolean algebra is isomorphic to the Boolean algebra of sets. The theorem is equivalent to the maximal ideal theorem, which can be proved without using the axiom of ...

A regular local ring is a local ring R with maximal ideal m so that m can be generated with exactly d elements where d is the Krull dimension of the ring R. Equivalently, R ...