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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 ...

Let pi be a unitary representation of a group G on a separable Hilbert space, and let R(pi) be the smallest weakly closed algebra of bounded linear operators containing all ...

An irreducible algebraic integer which has the property that, if it divides the product of two algebraic integers, then it divides at least one of the factors. 1 and -1 are ...

When the group order h of a finite group is a prime number, there is only one possible group of group order h. Furthermore, the group is cyclic.

A ring for which the product of any pair of ideals is zero only if one of the two ideals is zero. All simple rings are prime.

The prime subfield of a field F is the subfield of F generated by the multiplicative identity 1_F of F. It is isomorphic to either Q (if the field characteristic is 0), or ...

1 and -1 are the only integers which divide every integer. They are therefore called the prime units.

Given algebraic numbers a_1, ..., a_n it is always possible to find a single algebraic number b such that each of a_1, ..., a_n can be expressed as a polynomial in b with ...

An ideal I of a ring R is called principal if there is an element a of R such that I=aR={ar:r in R}. In other words, the ideal is generated by the element a. For example, the ...

Let alpha be a nonzero rational number alpha=+/-p_1^(alpha_1)p_2^(alpha_2)...p_L^(alpha_L), where p_1, ..., p_L are distinct primes, alpha_l in Z and alpha_l!=0. Then ...