Nature of Finitely Generated Ideals

I am a math Hobbyist reading Hungerford's book - Abstract Algebra (2nd Edition). In Chapter 6 on Ideals and Quotient Rings he writes (page 138 - see attached copies of pages 137 and 138 for context - whch is Finitely Generated Ideals):

"**Theorem 6.3** : Let R be a commutative ring with identity and , , ... ... , R.

Then the set I = { + + ... ... | , , ... ... , R} is an ideal in R"

I am trying to get a sense of the elements in the ideal in the Theorem above.

Are the elements always the sum of n non-zero elements and thus strictly of the form + + ... ... with no terms zero, or does the ideal contain (as I suspect) terms of the form with all other terms of the sum zero, + with all other terms zero, etc as well as the terms containing the sum of n elements.

Bernhard

Latex note: in the sum + I tried to make the subscripts 2 and 7 but for some reason Latex came up with an error every time I tried subscripts greater than 2! (Doh)

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Re: Nature of Finitely Generated Ideals

Forgot attachments - trying to add them.

Re: Nature of Finitely Generated Ideals

suppose a,b are typical elements in I, so that

.

then , so (I,+) is a subgroup of (R,+).

also, for any , so I is in fact an ideal.

indeed we can choose so each .

on the other hand, any ideal containing has to contain every R-linear combination of the ,

by closure of addition, and the multiplicative property of an ideal. so I is the smallest ideal containing the .

(it should be obvious that 0 terms are allowed. every ideal must contain 0, right?)

Re: Nature of Finitely Generated Ideals

Thanks Deveno

Working by myself I sometimes need the reasurrance of someone confirming these things. Thanks again. Most helpful.

Re: Nature of Finitely Generated Ideals

In his example on Page 138 Hungerford writes:

"In the ring Z[x], the ideal generated by the polynomial x and the constant polynomial 2 consists of all polynomials of the form:

f(x)x + g(x)2

It can be shown that this ideal is the ideal I of all polynomials with even constant term."

But if we take g(x) to be the zero polynomial we are left with polynomials of the form f(x)x (e.g (x + 1)x ) which do not have an even constant term. Am I missing something. {this example started my concern that no terms in the sum could be zero}

Bernhard

Re: Nature of Finitely Generated Ideals

0 (as a constant) is even.