Using number theoretic transform (NTT) is a means of performing efficient multiplication of large integers. The NTT notation of an integer is a vector. One can perform element-wise multiplication of two NTT vectors to compute an integer multiplication. As I've tested it is also possible to perform element-wise addition on vectors to compute integer addition. However, element-wise subtraction does not yield the correct result if any element become negative.
To be more precise, suppose $A$ and $B$ are two integers and $a = NTT(A)$ and $b=NTT(B)$. We know that $$A+B=INTT(a+b)$$ and $$A\times B=INTT(a \times b).$$ However, if any element gets negative in $a-b$, $$A-B\neq INTT(a-b).$$
Is there any work around to solve the issue regarding subtraction?