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!