What is an element of a rng called which is not the product of any elements?

Question

Let $R$ be a non-unital ring. Let $F:R\times R\longrightarrow R$ be a function given by the formula $F(x,y)=xy.$ Let $r\not\in\operatorname{im}(F).$ Such elements can exists, for example $2\in 2\mathbb Z$ isn't a product. It seems to be a major difference between unital and non-unital rings. I'm only starting to study non-unital rings and I thought it would be a good idea to understand this phenomenon better first. But I don't know any terminology, whence my question. What is the name (if there is any) of an element such as $r?$ Is there always such an element in a ring that actually doesn't have a unity? If not, what is the name of a ring in which such an element exists?

And finally, where can I read about it?

Answer 1

Suppose you have an algebra $A$ with an augmentation $\epsilon : A \to k$. Let $I$ be the kernel of $\epsilon$. Then elements that can not be written as products, can be thought of as elements of $I/I^2$ and are called indecomposable.

I am sure there are plenty of books, but I don't know about any specific books.

Answer 2

Consider the rng of cofinitely zero infinituples over $C$ or $R$. I.e. the set of all infinituples that are entirely zero after a while. This doesnt have a multiplicative identity but every element can be expressed as a product of rng elements.

As for the terminology or References, I remain woefully ignorant.