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<I,a> is not principal ideal . my doubt is I is generated by I=<x> ? Set S is the closed under multiplication is S contains unit elements of the ring ?
It will b too lengthy in exam if u try it this way. The question simply asks the no. of idempotents in Z_(105) which is equal to 2^k, where k is the no. prime divisors of 105 which is 3. So the answer is 2^3=8
Your formula does not valid in general as in Z_6, there are 2 elements (1 and 5) satisfying x^2=1. Here, the number of prime divisors of 6 is 2 (2 and 3). So, according to your statement, the no of elements in Z_6 satisfying x^2=1 is 2^2= 4 , which is not so.. Also check the definition of idempotent element: An element x in a ring R is said to be idempotent if x^2=x. (Here, x^2=1)
I have doubt codomain X with discrete topology f:R to X . there exist continuous function? if domain with discrete topology then any function is continuous
If R is endowed with usual topology, and X is endowed with discrete topology, then any constant function f:R-->X is continous, but there are so many funtion from R-->X which are not continuous.
Dear Pooja, I encourage you to take a look at the 2nd chapter ('FLOWS ON THE LINE') of the book 'NONLINEAR DYNAMICS AND CHAOS' by Steven H. Strogatz. I'm confident it will help clarify the concept for you. If you encounter any difficulties in understanding, feel free to comment here. We can arrange a class to discuss this topic further.
