Case of A Principal Ideal Domain
Suppose that R is a (commutative) principal ideal domain and M is a finitely-generated R-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module M up to isomorphism. In particular, it claims that
where F is a free R-module of finite rank (depending only on M) and T(M) is the torsion submodule of M. As a corollary, any finitely-generated torsion-free module over R is free. This corollary does not hold for more general commutative domains, even for R = K, the ring of polynomials in two variables. For non-finitely generated modules, the above direct decomposition is not true. The torsion subgroup of an abelian group may not be a direct summand of it.
Read more about this topic: Torsion (algebra)
Famous quotes containing the words case of, case, principal, ideal and/or domain:
“It was a maxim with Mr. Brass that the habit of paying compliments kept a man’s tongue oiled without any expense; and that, as that useful member ought never to grow rusty or creak in turning on its hinges in the case of a practitioner of the law, in whom it should be always glib and easy, he lost few opportunities of improving himself by the utterance of handsome speeches and eulogistic expressions”
—Charles Dickens (1812–1870)
“The youngest of four sons, but not the youngest of the family!—you conceive the sort of negligence that creeps over even the kindest maternities, in such case ...”
—Walter Pater (1839–1894)
“The principal thing children are taught by hearing these lullabies is respect. They are taught to respect certain things in life and certain people. By giving respect, they hope to gain self-respect and through self-respect, they gain the respect of others. Self-respect is one of the qualities my people stress and try to nurture, and one of the controls an Indian has as he grows up. Once you lose your self-respect, you just go down.”
—Henry Old Coyote (20th century)
“Our ideal ... must be a language as clear as glass—the person looking out of the window knows there is glass there, but he is not concerned with it; what concerns him is what comes through from the other side.”
—Elizabeth Bowen (1899–1973)
“When it had long since outgrown his purely medical implications and become a world movement which penetrated into every field of science and every domain of the intellect: literature, the history of art, religion and prehistory; mythology, folklore, pedagogy, and what not.”
—Thomas Mann (1875–1955)