Formal Quotient Ring Construction
Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows:
- a ~ b if and only if a − b is in I.
Using the ideal properties, it is not difficult to check that ~ is a congruence relation. In case a ~ b, we say that a and b are congruent modulo I. The equivalence class of the element a in R is given by
- = a + I := { a + r : r in I }.
This equivalence class is also sometimes written as a mod I and called the "residue class of a modulo I".
The set of all such equivalence classes is denoted by R/I; it becomes a ring, the factor ring or quotient ring of R modulo I, if one defines
- (a + I) + (b + I) = (a + b) + I;
- (a + I)(b + I) = (a b) + I.
(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of R/I is (0 + I) = I, and the multiplicative identity is (1 + I).
The map p from R to R/I defined by p(a) = a + I is a surjective ring homomorphism, sometimes called the natural quotient map or the canonical homomorphism.
Read more about this topic: Quotient Ring
Famous quotes containing the words formal, ring and/or construction:
“Good gentlemen, look fresh and merrily.
Let not our looks put on our purposes,
But bear it as our Roman actors do,
With untired spirits and formal constancy.”
—William Shakespeare (1564–1616)
“He will not idly dance at his work who has wood to cut and cord before nightfall in the short days of winter; but every stroke will be husbanded, and ring soberly through the wood; and so will the strokes of that scholar’s pen, which at evening record the story of the day, ring soberly, yet cheerily, on the ear of the reader, long after the echoes of his axe have died away.”
—Henry David Thoreau (1817–1862)
“There’s no art
To find the mind’s construction in the face:
He was a gentleman on whom I built
An absolute trust.”
—William Shakespeare (1564–1616)