Thread: Nature of Finitely Generated Ideals

1. 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 $\displaystyle c_{1}$, $\displaystyle c_{2}$, ... ... , $\displaystyle c_{n}$ $\displaystyle \in$ R.

Then the set I = {$\displaystyle r_{1}$$\displaystyle c_{1} + \displaystyle r_{2}$$\displaystyle c_{2}$ + ... ... $\displaystyle r_{n}$$\displaystyle c_{n} | \displaystyle r_{1}, \displaystyle r_{2}, ... ... \displaystyle r_{n}, \displaystyle \in R} is an ideal in R" I am trying to get a sense of the elements in the ideal in the Theorem above. Are the elements always the sum of n non-zero elements and thus strictly of the form \displaystyle r_{1}$$\displaystyle c_{1}$ + $\displaystyle r_{2}$$\displaystyle c_{2} + ... ... \displaystyle r_{n}$$\displaystyle c_{n}$ with no terms zero, or does the ideal contain (as I suspect) terms of the form $\displaystyle r_{1}$$\displaystyle c_{1} with all other terms of the sum zero, \displaystyle r_{2}$$\displaystyle c_{2}$ + $\displaystyle r_{1}$$\displaystyle c_{1} with all other terms zero, etc as well as the terms containing the sum of n elements. Bernhard Latex note: in the sum \displaystyle r_{2}$$\displaystyle c_{2}$ + $\displaystyle r_{1}$$\displaystyle c_{1}$ 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!

2. Re: Nature of Finitely Generated Ideals

Forgot attachments - trying to add them.

3. Re: Nature of Finitely Generated Ideals

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

$\displaystyle a = r_1c_1 + r_2c_2 +\dots + r_nc_n, b = s_1c_1 + s_2c_2 +\dots + s_nc_n$.

then $\displaystyle a-b = (r_1-s_1)c_1 + (r_2-s_2)c_2 +\dots + (r_n-s_n)c_n \in I$, so (I,+) is a subgroup of (R,+).

also, for any $\displaystyle d \in R, da = (dr_1)c_1 + (dr_2)c_2 +\dots + (dr_n)c_n \in I$, so I is in fact an ideal.

indeed we can choose $\displaystyle r_i = 1, r_j = 0, i \neq j$ so each $\displaystyle c_i \in I$.

on the other hand, any ideal containing $\displaystyle \{c_1,c_2,\dots ,c_n\}$ has to contain every R-linear combination of the $\displaystyle c_i$,

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

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

4. 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.

5. 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

6. Re: Nature of Finitely Generated Ideals

0 (as a constant) is even.