factor ideal

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English

Noun

factor ideal (plural factor ideals)

  1. (algebra, ring theory) For given ring R, any ideal I such that Q = R / I, the set of cosets of elements of I in R, is a ring (the quotient ring of I in R).
    • 1965 , Seth Warner, Modern Algebra, 1990, Dover, page 651,
      Consequently, we may speak of the sequence of invariant factor ideals of a finitely generated unitary module over a principal ideal domain A.
    • 1985 , Nathan Jacobson, Basic Algebra I, 2nd Edition, Dover, 2009, page 193,
      We shall now call the sequence of order ideals, ann z1, ann z2, ..., whose uniqueness has just been proved, the invariant factor ideals of the module M.
    • 2012, Arthur Knoebel, Sheaves of Algebras over Boolean Spaces, Springer (Birkhäeser), page 246:
      Recall that a factor ideal I of a half-shell A comes from a factor congruence θ of A where I = 0/θ. Factor ideals and congruences will be needed whenever we want to show that we have embedded a half-shell into a Baer–Stone half-shell. []
      2.1 Definition. Remember that a half-shell A is called a Baer–Stone half-shell if the right and left annihilators of every single element of A are factor ideals. It is called finitely Baer-Stone if the annihilators on both sides of any finite subset of A are factor ideals.