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A nonzero ring S whose only (two-sided) ideals are S itself and zero. Every commutative simple ring is a field. Every simple ring is a prime ring.

Let P(L) be the set of all prime ideals of L, and define r(a)={P|a not in P}. Then the Stone space of L is the topological space defined on P(L) by postulating that the sets ...

A Lie algebra over a field of characteristic zero is called semisimple if its Killing form is nondegenerate. The following properties can be proved equivalent for a ...

Let R be a ring. If phi:R->S is a ring homomorphism, then Ker(phi) is an ideal of R, phi(R) is a subring of S, and R/Ker(phi)=phi(R).

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 ...

A unique factorization domain, called UFD for short, is any integral domain in which every nonzero noninvertible element has a unique factorization, i.e., an essentially ...

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

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 ...