  In this paper we study arithmetic computations in the nonassociative, and
noncommutative free polynomial ring $\mathbb{F}\{x_1,x_2,\ldots,x_n\}$. Prior
to this work, nonassociative arithmetic computation was considered by Hrubes,
Wigderson, and Yehudayoff [HWY10], and they showed lower bounds and proved
completeness results. We consider Polynomial Identity Testing (PIT) and
polynomial factorization over $\mathbb{F}\{x_1,x_2,\ldots,x_n\}$ and show the
following results. (1) Given an arithmetic circuit $C$ of size $s$ computing a
polynomial $f\in \mathbb{F} \{x_1,x_2,\ldots,x_n\}$ of degree $d$, we give a
deterministic $poly(n,s,d)$ algorithm to decide if $f$ is identically zero
polynomial or not. Our result is obtained by a suitable adaptation of the PIT
algorithm of Raz-Shpilka [RS05] for noncommutative ABPs. (2) Given an
arithmetic circuit $C$ of size $s$ computing a polynomial $f\in \mathbb{F}
\{x_1,x_2,\ldots,x_n\}$ of degree $d$, we give an efficient deterministic
algorithm to compute circuits for the irreducible factors of $f$ in time
$poly(n,s,d)$ when $\mathbb{F}=\mathbb{Q}$. Over finite fields of
characteristic $p$, our algorithm runs in time $poly(n,s,d,p)$.
