R/I as a bimodule - Tensor Products - D&F page 366 - Example 2 - R/I as a bimodule

Peter

Well-known member
MHB Site Helper
I am reading Dummit and Foote Section 10.4: Tensor Products of Modules. I would appreciate some help in understanding Example 2 on page 366 concerning viewing the quotient ring $$\displaystyle R/I$$ as an $$\displaystyle (R/I, R)$$-bimodule.

Example (2) D&F page 366 reads as follows:

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"(2) Let I be an ideal (two sided) in the ring $$\displaystyle R$$. Then the quotient ring $$\displaystyle R/I$$ is an $$\displaystyle (R/I, R)$$-bimodule. ... ... ...

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Now for $$\displaystyle R/I$$ to be a $$\displaystyle (R/I, R)$$-bimodule we require that :

1. $$\displaystyle R/I$$ is a left $$\displaystyle R/I$$-module

2. $$\displaystyle R/I$$ is a right $$\displaystyle R$$-module

3. (a + I) ( (b+ I) r) = ( (a + I) (b+ I) ) r where a+I, b+I belong to R/I and r is in R.

I have problems with the meaning and rules governing operations on elements in 2 above, and a similar problem with the operations in 3.

Consider now, $$\displaystyle R/I$$ as a right $$\displaystyle R$$-module

Following Dummit and Foote's definition of a module on page 337 (see attachment) and following the definition closely and carefully (and adjusting for a right module rather than a left module), for $$\displaystyle M = R/I$$ to be a right $$\displaystyle R$$-module we require

(1) $$\displaystyle R/I$$to be an abelian group under the operation +, which is achieved under the normal definition of addition of cosets, visually:

$$\displaystyle (a + I) + (b + I) = (a+b) + I$$

(2) an action of $$\displaystyle R$$ on $$\displaystyle R/I$$ (that is a map $$\displaystyle R/I \times R \to R/I$$) denoted by $$\displaystyle ( a + I ) r$$ for all $$\displaystyle (a + I) \in R/I \text{ and for all } r \in R$$ which satisfies:

(a) $$\displaystyle ( a + I ) ( r + s) = (a + I) r + (a + I) s \text{ where } (a + I) \in R/I \text{ and } r, s \in R$$

... ... and so on for conditions (b), (c) and (d) - see D&F page 337 (see attachment)

My question is as follows:

How do we interpret the action $$\displaystyle ( a + I ) r$$ , and also how do we interpret, indeed form/calculate expressions like $$\displaystyle (a + I) r$$ in expressions (a) above ... also actually in (b), (c), (d) as well

I would appreciate some help.

Peter

Last edited:

Deveno

Well-known member
MHB Math Scholar
It is natural to set:

$(a + I)r = ar + I$

(since $I$ is a 2-sided ideal, $Ir = I$).