Abstruse algebra Basic domains

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 10 June 15:51   

    Motivation: The abstraction of divisibility is axial to the abstraction of ring theory. Basic domains are a advantageous apparatus for belief the altitude beneath which concepts like divisibility and different factorization are well-behaved.

    Definition An basic area is a capricious ring R with 1_R
eq 0_R such that for all a, b in R, the account ab=0 implies either a = 0 or b = 0.

    An agnate analogue is as follows:

    Definition Accustomed a ring R, a zero-divisor is an aspect a in R such that exists x in R, x
eq 0 such that a

    Definition An basic area is a capricious ring R with 1_R
eq 0_R and with no non-zero zero-divisors.

    Remark An basic area has a advantageous abandoning property: Let R be an basic area and

    let a, b, c in R with a
eq 0 . Then a b = a c implies b = c. For this

    reason an basic area is sometimes alleged a abandoning ring.


    #The set mathbb of integers beneath accession and multiplication is an basic domain. However, it is not a acreage back the aspect 2 in has no multiplicative inverse.

    #The set atomic ring is not an basic area back it does not amuse 0
eq 1.

    #The set mathbb_6 of accordance classes of the integers modulo 6 is not an basic area because [2]

    Theorem: Any acreage F is an basic domain.

    Proof: Accept that F is a acreage and let a in F, a
eq 0. If ax = 0 for some x in F, then accumulate by a^ to see that ax = 0 Rightarrow a^(ax) = a^0 Rightarrow 1x = 0 Rightarrow x = 0. F cannot, therefore, accommodate any aught divisors. Thus, F is an basic domain.square

    Definition If R is a ring, then the set of polynomials in admiral of x with coefficients from


    is aswell a ring, alleged the polynomial ring of R and accounting R[x]. Anniversary such polynomial is a bound sum of

    terms, anniversary appellation getting of the anatomy r x^n area r in R and x^n represents the

    n-th ability of x. The arch appellation of a polynomial is authentic as that appellation of the polynomial which contains

    the accomplished ability of x in the polynomial.

    Remark A polynomial equals 0 if and alone if anniversary of its coefficients equals 0.

    Theorem: Let R be an basic area and let R[x] be the ring of polynomials in admiral of


    whose coefficients are elements of R. Then R[x] is an basic area if and alone if R is.

    Proof If capricious ring R is not an basic domain, it contains two non-zero elements a and

    b such that a b = 0. Then the polynomials a x and b x are non-zero

    elements of R[x] and a x b x = a b x x = 0 x x = 0 . Appropriately if R is not an integral

    domain, neither is R[x].

    Now let R be an basic area and let A and B be polynomials in R[x].

    If the polynomials are both non-zero, then anniversary one has a non-zero arch term, alarm them a x^m and

    b x^n. That these are the arch agreement of polynomials A and B agency that the leading

    term of the artefact A B of these polynomials is a b x^. Back R is an basic domain

    and a, b in R, a b
eq 0. This means, by the Acknowledgment above, that the artefact A B is not

    zero either. This agency that R[x] is an basic domain.



Tags: abstract, field, domain, domains

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