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quasar987 said:You cannot "cancel" f(x) because in a ring, elements do not in general have multiplicative inverses.
But consider the following. Since f is nontrivial, there exists x in R such that f(x) [itex]\neq[/itex]0. Then f(x)=f(x)f(1) and substracting f(x), we get 0=f(x)(1-f(1)).
And now consider the two possible cases: 1-f(1)=0 and 1-f(1)[itex]\neq[/itex]0.
hkhk said:f(1R)= f(1R)f(1R)
so
f(1R)- 1s.f(1R) = 0
but when you write this are you not already assuming that f(1R) = 1s which is what we are trying to prove
hkhk said:yes but this proof does not use the given property that it is a nontrivial homomorphism
hkhk said:yes but this proof does not use the given property that it is a nontrivial homomorphism,
there is some x in R for f(x) /= 0s
i feel like this is how we should start
f(x)1s= f(x) = f (x1R)= f(x)f(1R)
f(x) =f(x)f(1R)
f(x) - f(x)f(1R) =0
hkhk said:ok, thanks !.
so it is ok to start this proof by saying
consider 1R=1R.1R then,
(the steps in post 6 are correct)
The equation 1R=1R.1R is a mathematical expression that represents the concept of equality. It means that one unit of a quantity (1R) is equal to one unit of the same quantity (1R) multiplied by another unit of the quantity (1R).
Yes, the equation 1R=1R.1R is true as it follows the fundamental mathematical principle of equality. Any quantity is always equal to itself multiplied by one, which is represented by the 1R.1R term in this equation.
No, the equation 1R=1R.1R cannot be simplified any further as it is already in its simplest form. The equation represents the concept of equality and cannot be simplified any further without changing its meaning.
The equation 1R=1R.1R is commonly used in science to represent the concept of conservation of mass or matter. It states that the total mass of a closed system remains constant, and matter cannot be created or destroyed, only transformed.
The equation 1R=1R.1R is used in various real-life situations, such as in chemistry to balance chemical reactions, in physics to calculate the momentum of objects, and in engineering to design proportional systems. It is also used in everyday life, for example, when measuring ingredients for a recipe or calculating the distance traveled by a car at a constant speed.