Please explain what you are talking about! What are f and g? Are they members of R?regularngon said:Homework Statement
Let R be a commutative ring and a,b in R. Show that the canonical image of ab in R/(f - f^2*g) is idempotent. Give an example where this idempotent is not 0 or 1.
Homework Equations
None.
The Attempt at a Solution
Well I've tried playing with the properties of ideals such as multiplicative closure under R but I've had no luck.
Ring theory is a branch of abstract algebra that studies the mathematical structures called rings. These structures consist of a set of elements and two binary operations, addition and multiplication, that satisfy certain properties.
In ring theory, idempotence is a property that some elements in a ring may have. An element a is idempotent if a*a = a, meaning that when the element is multiplied by itself, it remains unchanged.
To show idempotence in R/(f-f^2*g), we need to prove that for any element x in the ring, (x+x)*(x+x) = x+x. This can be done by using the definition of the quotient ring and the properties of the element x, f, and g.
One example of idempotence in R/(f-f^2*g) is the element (f+g). When multiplied by itself, (f+g)*(f+g) = f^2 + 2fg + g^2. Since f^2 and g^2 are both equal to 0 in this ring, this simplifies to (f+g)*(f+g) = 2fg = f+g. Therefore, (f+g) is an idempotent element in R/(f-f^2*g).
Idempotence in R/(f-f^2*g) is significant because it allows us to simplify calculations and make certain proofs easier. It also helps us to better understand the structure of the ring and its elements.