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On page 6 in 1.11 Lemma, we have the following:

**[see attachment]**"Let S be a subring of the ring R, and let [TEX] \Gamma [/TEX] be a subset of R.

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

Thus, [TEX] S[ \Gamma ] [/TEX] is a subring of R which contains both S and [TEX] \Gamma [/TEX], 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 [TEX] \Gamma [/TEX].

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

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

[TEX] S[ \alpha ] = \{ \ {\sum}_{i = 0}^{t} s_i \alpha^i : t \in {\mathbb{N}}_0 \ s_0, s_1, ... ... , s_t \in S \} [/TEX] ......... (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 [TEX] a_1, a_2, ... ... , a_n \in R [/TEX] and use an indeterminate x (whatever that is?) to form sums like the following:

[TEX] a_n x^n + a_{n-1} + ... ... + a_1x + a_0 [/TEX] .......... (2)My problems are as follows:

(a) It looks like (1) and (2) have the same structure BUT [TEX] \alpha [/TEX] 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 [TEX] a_1, a_2, ... ... , a_n \in R [/TEX] is just a subset of R - whereas [TEX] s_0, s_1, ... ... , s_t [/TEX] 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]