The discussion focuses on demonstrating the idempotence of the canonical image of the product ab in the quotient ring R/(f - f^2*g), where R is a commutative ring and a, b are elements of R. Participants emphasize the need to explore the relationships between the elements a, b, f, and g to establish idempotence. An example is requested where the resulting idempotent is neither 0 nor 1, highlighting the complexity of the problem. The conversation reveals a gap in understanding the roles of f and g within the context of the ring.
PREREQUISITESMathematics students, particularly those studying abstract algebra, ring theory enthusiasts, and educators seeking to deepen their understanding of idempotent elements in commutative rings.
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.