This is what I have for the definition of kernel; The kernel of $f$, denoted by $ker f$, is the set of elements in $R$ annihilated by $f$: $ker f = \{a \in R : f (a) = 0\}$.
and the definition of annihilator; $ann(a)=\{x \in R: xa=0\}$
is the kernel the same thing as annihilator?
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abstract-algebra
discrete-mathematics
2 Answers
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No, they are not "the same thing," although sometimes they coincide.
The term "kernel" is used with respect to homomorphisms, whereas you can talk about the annihilator of any subset of a module.
Of course there is some overlap since sets of homomorphisms can be modules, and then the two might be defining the same thing.
If, for example, you let the elements of $R$ act by right multiplication on $R$, then this produces left $R$ linear homomorphisms $R\to R$. As you can see, the kernel of the homomorphism $f:x\mapsto xr$ for some fixed $r$ is exactly the left annihilator of $r$.
Suppose you have a homomorphism $f$ of a module $M$: the kernel is as you described: $\{m\in M\mid f(m)=0\}$. Obviously the thing we are quantifying over is not from a ring, so it does not fit the above definition, although it is definitely called the kernel.
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Example: if $R$ is an algebra, $a \in A$ and if $f:A \to A$ is defined by $f(x)=xa$, then $f$ is linear and $ker(f)=ann(a)$.