Abstruse algebra Basic domains

 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.

    Examples:

    #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

    $R$

    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

    $x$

    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.