I am confused by $A[x]$, I can't seem to grasp the concept. I saw this in a book and the proof was left as an exercise.

Any help would be much appreciated.

I know that if $B$ is a subring of $A$, then $B$ is closed under multiplication, addition and negatives.

In any ring $A, A[x]$ is the set of all the polynomials in $x$ whose coefficients are in $A$, with addition and multiplication of polynomials. But what is meant by "all the polynomials in $x$" ?


1 Answer 1


$A[x]$ is the set of all formal expressions of the form $\sum_{k=0}^\infty a_kx^x$ where the $a_k$'s are elements of the ring $A$ and $a_k=0$ for all $k$ except a finite number of indexes. Every element of $A[x]$ is expressed usually with a symbol $symbol(x)$. It is only notation.

For example, $p(x)=1+x^2+2016x^5$ can be written on the form $p(x)=\sum_{k=0}^\infty a_kx^k$, where $a_0=1$, $a_2=1$, $a_5=2016$ and $a_k=0$ for $k\neq 0,2,5$.

From now,if $B$ is a subring of $A$,then $B[x]$ is a subring of $A[x]$ if $B[x]$ is a ring and $B[x]\subseteq A[x]$. But both of this conditions are easly to prove.


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