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Integral domainIn abstract algebra, an integral domain, is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility.Alternatively and equivalently, integral domains may be defined as commutative rings in which the zero ideal {0} is prime, or as the subrings of fields.
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The prototypical example is the ring Z of all integers.
Every field is an integral domain. Conversely, every Artinian integral domain is a field. In particular, the only finite integral domains are the finite fields.
Rings of polynomials are integral domains if the coefficients come from an integral domain. For instance, the ring Z[X] of all polynomials in one variable with integer coefficients is an integral domain; so is the ring R[X,Y] of all polynomials in two variables with real coefficients .
The set of all real numbers of the form a + b√2 with a and b integers is a subring of R and hence an integral domain. A similar example is given by the complex numbers of the form a + bi with a and b integers (the Gaussian integers).
If U is a connected open subset of the complex number plane C, then the ring H(U) consisting of all holomorphic functions f : U -> C is an integral domain. The same is true for rings of analytical functions on connected open subsets of analytical manifolds.
If R is a commutative ring and P is a prime ideal in R, then the factor ring R/P is an integral domain.
If a and b are elements of the integral domain R, we say that a divides b or a is a divisor of b or b is a multiple of a if and only if there exists an element x in R such that ax = b.
If a divides b and b divides c, then a divides c. If a divides b, then a divides every multiple of b. If a divides two elements, then a also divides their sum and difference.
The elements which divide 1 are called the units of R; these are precisely the invertible elements in R. Units divide all other elements.
If a divides b and b divides a, then we say a and b are associated elements. a and b are associated if and only if there exists a unit u such that au = b.
If q is a non-unit, we say that q is an irreducible element if q cannot be written as a product of two non-units.
If p is a non-zero non-unit, we say that p is a prime element if, whenever p divides a product ab, then p divides either a or b.
This generalizes the ordinary definition of prime number in the ring Z, except that it allows for negative prime elements. If p is a prime element, then the principal ideal (p) generated by p is a prime ideal. Every prime element is irreducible (here, for the first time, we need R to be an integral domain), but the converse is only true in a unique factorization domain.
If R is a given integral domain, the smallest field Quot(R) containing R as a subring is uniquely determined up to isomorphism and is called the field of fractions or quotient field of R. It consists of all fractions a/b with a and b in R and b ≠ 0. The field of fractions of the integers is the field of rational numbers. The field of fractions of a field is that field itself.
The characteristic of every integral domain is either zero or a prime number.
If R is an integral domain with prime characteristic p, then f(x) = xp defines an injective ring homomorphism f : R -> R, the Frobenius homomorphism.
And few that will carry their looks or their truth to the grave.
I wish one could know them, I wish there were tokens to tell
And then one could talk with them friendly and wish them farewell
And brushing your elbow unguessed-at and not to be told
The lads that will die in their glory and never be old.
XXIV
Say, lad, have you things to do?
Quick, and if 'tis work for two,
call.html">Call me, I shall hear you call;
Where a man.html">man's no use at all;
Ere the wholesome flesh decay,
And the lips lack breath to say,
When Fred and I would meet,
We fought and I was beat.
So then the summer fields about,
Rose Harland on her Sundays out
Though now 'tis not with Fred:
Is worth a dozen dead.
Fred keeps the house all kinds of weather,
When Rose and I walk out together
A year ago, my love.html">love and I,
Was talking to itself alone.
A country lover and his lass;
And time shall put them both to bed.html">bed,
And he beside another love."
And sure enough beneath the tree
And overhead the aspen heaves
And I spell nothing in their stir,
And plain for her to understand
When I shall sleep with clover clad,
That I was used to drive
When I was man alive?"
Ay, the horses trample,
No change though you lie.html">lie under
Along the river shore,
Now I stand up no more?"
Ay, the ball is flying,
The goal stands up, the keeper
That I thought hard to leave,
As she lies.html">lies down at eve?"
Ay, she lies down lightly,
Your girl is well contented.
Now I am thin and pine,
A better bed than mine?"
Yes, lad, I lie easy,
.
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