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A primitive subgroup of the symmetric group S_n is equal to either the alternating group A_n or S_n whenever it contains at least one permutation which is a q-cycle for some ...

A positive integer n is kth powerfree if there is no number d such that d^k|n (d^k divides n), i.e., there are no kth powers or higher in the prime factorization of n. A ...

A primary ideal is an ideal I such that if ab in I, then either a in I or b^m in I for some m>0. Prime ideals are always primary. A primary decomposition expresses any ideal ...

Given an ideal A, a semiprime ring is one for which A^n=0 implies A=0 for any positive n. Every prime ring is semiprime.

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 finite field is a field with a finite field order (i.e., number of elements), also called a Galois field. The order of a finite field is always a prime or a power of a ...

A number is squareful, also called nonsquarefree, if it contains at least one square in its prime factorization. The first few are 4, 8, 9, 12, 16, 18, 20, 24, 25, ... (OEIS ...

A primefree sequence is sequence whose terms are never prime. Graham (1964) proved that there exist relatively prime positive integers a and b such that the recurrence ...

Every sufficiently large odd number is a sum of three primes (Vinogradov 1937). Ramachandra and Sankaranarayanan (1997) have shown that for sufficiently large n, the error ...