Understanding Ring Embeddings and Irreducible Polynomials: Questions Answered

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Discussion Overview

The discussion revolves around the concepts of ring embeddings and irreducible polynomials within the context of ring theory. Participants explore the definitions and implications of isomorphisms, homomorphisms, and the nature of ideals in polynomial rings, particularly in relation to irreducibility over specific fields.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the definition of an isomorphism and its implications, suggesting that if R is isomorphic to R1, then R1 should also be isomorphic to R, leading to confusion about the nature of embeddings.
  • Another participant challenges the terminology used in the original post, suggesting that the term "isomorphism" may have been misapplied and proposes "homomorphism" as a more accurate term.
  • A participant clarifies that an ideal (p(x)) in F[x] represents the set of all multiples of the polynomial p(x), and discusses the nature of irreducibility in relation to maximal ideals.
  • There is a discussion about the interpretation of -9 in the context of integers mod 31, with one participant explaining that it represents the additive inverse of 9, which is 22 mod 31.
  • Another participant emphasizes the distinction between "onto" and "into" in the definitions provided by the book, which may contribute to the confusion regarding ring embeddings.

Areas of Agreement / Disagreement

Participants express differing interpretations of the definitions related to isomorphisms and embeddings, indicating a lack of consensus on the terminology and its implications. The discussion remains unresolved regarding the clarity of these concepts.

Contextual Notes

There are potential limitations in understanding due to the nuances in definitions of isomorphisms and homomorphisms, as well as the specific nature of ideals in polynomial rings. The discussion also highlights the importance of precise language in mathematical definitions.

buzzmath
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In my book it says that a Ring R can be imbedded in a ring R1 if there is an isomorphism of R onto R1 and we call R1 the over-ring. Some of the things that the author goes on to talk about makes me think that R can just be fit into R1 almost like it's a subset (which I know it's not) of R1. What I'm wondering is that if R is isomorphic to R1 then isn't R1 isomorphic to R so if R can be imbedded in R1 then R1 can be imbedded into R? Can anyone help to make this a little clearer to me?

Another question I had has to do with polynomial rings. If you're trying to show that a certain polynomial is irreducible over a field F[x] is to show that the ideal A = (p(x)) in F[x] is a maximal ideal? What exactly does an ideal (p(x)) in F[x] look like? An example practice problem I have is show that x^3-9 is irreducible over the integers mod31. So I need to show that the ideal (x^3-9 ) is a maximal ideal. what does an ideal of this form look like. maybe not just this example but in general. Also, where does the -9 come in since the integers mod31 are {0,1,2,...,30} which are all positive.

thanks for any help
 
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All I can suggest is that you reread your book more closely since what you've written is obviously wrong: an 'isomorphism ... onto'? I don't think you meant to write that.

-9 is just "minus 9", the additive inverse of 9, so that is obviously 22, mod 31, since 9+22=31 which is zero mod 31. The things mod 31 are not 'just' the numbers 0,...,30. Those labels are good choices of elements of equivalence classes of elements, with addition of the equivalence classes as the operation.
 
Are you sure your book didn't say "homomorphism"? If R is isomorphic to R1 then R is not a subring of R1, it is essentially the same as R1.
 
My book defines an isomorphism as a one-to-one homomorphism of R into R1. Then it says that two rings are isomorphic if there is an isomorphism of one onto the other. Then it says a ring R can be imbedded in a ring R1 if there is an isomrphism of R into R1.
 
That is an injective homomorphism. It is *not* an isomorphism.
 
buzzmath said:
My book defines an isomorphism as a one-to-one homomorphism of R into R1. Then it says that two rings are isomorphic if there is an isomorphism of one onto the other. Then it says a ring R can be imbedded in a ring R1 if there is an isomrphism of R into R1.

Notice that one says "onto" and the other "into".
 

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