Results 1 to 1 of 1

Math Help - Rings of the form R[X] - Ring Adjunction

  1. #1
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    559
    Thanks
    2

    Rings of the form R[X] - Ring Adjunction

    I am reading R.Y Sharp's book: "Steps in Commutative Algebra".

    On page 6 in 1.11 Lemma, we have the following: [see attachment]

    "Let S be a subring of the ring R, and let  \Gamma be a subset of R.

    Then  S[ \Gamma ] is defined as the intersection of all subrings of R which contain S and  \Gamma .

    Thus,  S[ \Gamma ] is a subring of R which contains both S and  \Gamma , and it is the smallest such subring of R in the sense that it is contained in every other subring of R that contains S and  \Gamma .

    In the special case in which  \Gamma is a finite set  \{ \alpha_1, \alpha_2, ... ... ,  \alpha_n \} we write  S[ \Gamma ] as  S [ \alpha_1, \alpha_2, ... ... ,  \alpha_n ] .

    In the special case in which S is commutative, and  \alpha  \in R is such that  \alpha s = s \alpha for all  s \in S we have

     S[ \alpha ] = \{ \   {\sum}_{i = 0}^{t} s_i \alpha^i : t \in {\mathbb{N}}_0 \  s_0, s_1, ... ... , s_t \in S   \} .............................................. (1)


    ------------------------------------------------------------------------------------------------------------------------------------

    Then on page 7 Sharp writes:

    Note that when R is a commutative ring and X is an indeterminate, then it follows from 1.11 Lemma that our earlier use of R[X] to denote the polynomial ring is consistent with this new use of R[X] to denote 'ring adjunction'.

    -------------------------------------------------------------------------------------------------------------------------------------

    Now in the polynomial ring R[X] we take a subset of ring elements  a_1, a_2, ... ... , a_n \in R and use an indeterminate x (whatever that is?) to form sums like the following:

     a_n x^n + a_{n-1} + ... ... + a_1x + a_0  .................................................. ...... (2)


    My problems are as follows:

    (a) It looks like (1) and (2) have the same structure BUT  \alpha is a member of the ring R, and also the subring S whereas x is not a member of R but is an "indeterminate" [maybe I am overthinking this and it does not matter??] Can someone please clarify this matter?

    (b) Again, (1) and (2) seem to have the same structure BUT  a_1, a_2, ... ... , a_n \in R is just a subset of R - whereas  s_0, s_1, ... ... , s_t  are elements of a subring. Does this matter? Can someone please clarify?

    (c) Sharp specifies that S has to be commutative - but why? I cannot see how this is needed in his Proof on the bottom of page 6. Can someone help.

    I would be grateful if someone can clarify the above.

    Peter
    Last edited by Bernhard; August 23rd 2013 at 08:11 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Are all ideals of a ring rings themselves? A question
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 17th 2013, 11:43 AM
  2. Ring based on families of integers - The Function Ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 8th 2013, 09:17 AM
  3. Replies: 5
    Last Post: December 28th 2012, 11:27 PM
  4. Ring theory, graded rings and noetherian rings
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: January 4th 2012, 12:46 PM
  5. a book on semigroup rings and group rings
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 2nd 2011, 05:35 AM

Search Tags


/mathhelpforum @mathhelpforum