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After defining an R-algebra at the bottom of page 5 (see attachment from Sharp), on the top of page 6 we find the following:
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"We should point out at once that the concept of an R-algebra introduced in 1.9 above occurs very frequently in ring theory, simply because every ring is a Z-algebra. We explain in 1,10 why this is the case."
-----------------------------------------------------------------------------------Sharp then proceeds as follows:-----------------------------------------------------------------------------------
"Let $$ R $$ be a ring. Then the mapping $$ F \ : \ \mathbb{Z} \to R $$ defined by $$ f(n) = n(1_R) $$ for all $$ n \in \mathbb{Z} $$ is a ring homomorphism and, in fact, is the only ring homomorphism from $$ \mathbb{Z} $$ to $$ R $$.
Here
$$ n(1_R) = 1_R + 1_R + ... \ ... + 1_R $$ (n-terms) ... for n > 0
$$ n(1_R) = 0_R $$ for n = 0
and
$$ n(1_R) = (-1_R) + (-1_R) + ... \ ... + (-1_R) $$ (n-terms) ... for n < 0
It should be clear from 1.5 (The subring criterion - see attachment) that the intersection of any non-empty family of subrings of a ring R is again a subring of R, This observation leads to the following Lemma (Lemma 1.10 - see attachment, page 6). ... ... ... "
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My first problem with the above is this:
Sharp writes: "Let $$ R $$ be a ring. Then the mapping $$ F \ : \ \mathbb{Z} \to R $$ defined by $$ f(n) = n(1_R) $$ for all $$ n \in \mathbb{Z} $$ is a ring homomorphism and, in fact, is the only ring homomorphism from $$ \mathbb{Z} $$ to $$ R $$."
But why is this the only ring homomorphism from $$ \mathbb{Z} $$ to $$ R $$?
My second problem is as follows: Sharp connects establishing a Z-Algebra (section 1,10 page 6) with establishing the structure of a polynomial ring - but what is the relationship and big picture here?
My third problem is the following:
Dummit and Foote on page 339 define Z-Modules (see attachment) , but seem to structure them slightly differently using a an element a from an abelian group (and not the identity of the group) whereas Sharp uses the multiplicative identity of a ring in establishing a Z-algebra. I presume this is because the abelian groups are being treated as additive and one cannot use the additive identity - can someone please confirm this please - or clarify and explain the links between Z_algebras and Z-modules.
I would really appreciate some help and clarification of these issues.
Peter
----------------------------------------------------------------------------------
"We should point out at once that the concept of an R-algebra introduced in 1.9 above occurs very frequently in ring theory, simply because every ring is a Z-algebra. We explain in 1,10 why this is the case."
-----------------------------------------------------------------------------------Sharp then proceeds as follows:-----------------------------------------------------------------------------------
"Let $$ R $$ be a ring. Then the mapping $$ F \ : \ \mathbb{Z} \to R $$ defined by $$ f(n) = n(1_R) $$ for all $$ n \in \mathbb{Z} $$ is a ring homomorphism and, in fact, is the only ring homomorphism from $$ \mathbb{Z} $$ to $$ R $$.
Here
$$ n(1_R) = 1_R + 1_R + ... \ ... + 1_R $$ (n-terms) ... for n > 0
$$ n(1_R) = 0_R $$ for n = 0
and
$$ n(1_R) = (-1_R) + (-1_R) + ... \ ... + (-1_R) $$ (n-terms) ... for n < 0
It should be clear from 1.5 (The subring criterion - see attachment) that the intersection of any non-empty family of subrings of a ring R is again a subring of R, This observation leads to the following Lemma (Lemma 1.10 - see attachment, page 6). ... ... ... "
------------------------------------------------------------------------------------
My first problem with the above is this:
Sharp writes: "Let $$ R $$ be a ring. Then the mapping $$ F \ : \ \mathbb{Z} \to R $$ defined by $$ f(n) = n(1_R) $$ for all $$ n \in \mathbb{Z} $$ is a ring homomorphism and, in fact, is the only ring homomorphism from $$ \mathbb{Z} $$ to $$ R $$."
But why is this the only ring homomorphism from $$ \mathbb{Z} $$ to $$ R $$?
My second problem is as follows: Sharp connects establishing a Z-Algebra (section 1,10 page 6) with establishing the structure of a polynomial ring - but what is the relationship and big picture here?
My third problem is the following:
Dummit and Foote on page 339 define Z-Modules (see attachment) , but seem to structure them slightly differently using a an element a from an abelian group (and not the identity of the group) whereas Sharp uses the multiplicative identity of a ring in establishing a Z-algebra. I presume this is because the abelian groups are being treated as additive and one cannot use the additive identity - can someone please confirm this please - or clarify and explain the links between Z_algebras and Z-modules.
I would really appreciate some help and clarification of these issues.
Peter