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The semigroup algebra K[S], where K is a field and S a semigroup, is formally defined in the same way as the group algebra K[G]. Similarly, a semigroup ring R[S] is a ...

A knot that secures a rope to a post, ring, another rope, etc., but does not keep its shape by itself.

Let R be a commutative ring. A category C is called an R-category if the Hom-sets of C are R-modules.

A ring that is commutative under multiplication, has a multiplicative identity element, and has no divisors of 0. The integers form an integral domain.

Let P be the set of prime ideals of a commutative ring A. Then an affine scheme is a technical mathematical object defined as the ring spectrum sigma(A) of P, regarded as a ...

A tree is planted at each lattice point in a circular orchard which has center at the origin and radius r. If the radius of trees exceeds 1/r units, one is unable to see out ...

A special ideal in a commutative ring R. The Jacobson radical is the intersection of the maximal ideals in R. It could be the zero ideal, as in the case of the integers.

Let R be a number ring of degree n with 2s imaginary embeddings. Then every ideal class of R contains an ideal J such that ||J||<=(n!)/(n^n)(4/pi)^ssqrt(|disc(R)|), where ...

A right or left ideal of a ring. The term is used especially in noncommutative rings to denote a right ideal that is not a left ideal, or conversely.

The double-struck capital letter I, I, is a symbol sometimes used instead of Z for the ring of integers. In contrast, the lower case symbol i is used to refer to the ...