Question about what I am allowed to do in a finite set which has the closure property under a certain operation

Question

I have a question regarding what I am allowed to do whenever I have a set of elements from a ring that has the closure property. Suppose we have a ring $(A, +, \cdot)$ and we consider the finite set $K = \{a_1, a_2, ..., a_n\} \subset A$ so that it has the closure property under the $\cdot$ operation. Would I be allowed to choose an element $a_i \in K, 1 \leq i \leq n$ and write that $K = \{a_ia_1, a_ia_2, ..., a_ia_n\}$ and also state that $\sum_{k=1}^n a_k = \sum_{k=1}^n a_ia_k$, where $a_i$ is the fixed element I chose?

Answer 1

A counterexample, even when adding the condition $0_A\notin K,$ is the following:

$$A=\Bbb R[X]/I\text{ where }I=(X^2-X),\quad K=\{1_A,X\bmod I\},\quad a_i=X\bmod I.$$

On the other hand, if $0_A\notin K$ and $A$ is an integral domain then the map $K\to K,\; x\mapsto a_i x$ will be injective, hence bijective since $K$ is finite.