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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'.
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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
[Note: This has also been posted on MHF]
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
[Note: This has also been posted on MHF]