Definition
Let G be a group, written multiplicatively, and let R be a ring. The group ring of G over R, which we will denote by R, is the set of mappings f : G → R of finite support, where the product of a scalar in R and a vector (or mapping) f is defined as the vector, and the sum of two vectors f and g is defined as the vector . To turn the commutative group R into a ring, we define the product of f and g to be the vector
The summation is legitimate because f and g are of finite support, and the ring axioms are readily verified.
Some variations in the notation and terminology are in use. In particular, the mappings such as f : G → R are sometimes written as what are called "formal linear combinations of elements of G, with coefficients in R":
or simply
where this doesn't cause confusion.
Read more about this topic: Group Ring
Famous quotes containing the word definition:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)