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

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?

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What is a "rng"? Google comes up with a "random number generator"; but this doesn't seem to be the idea here. –  Christian Blatter Jan 28 '12 at 19:08
"Rng" is a fancy term for a non-unital ring. –  user23211 Jan 28 '12 at 19:13
When there ever was an abuse of language: Here is one. –  Christian Blatter Jan 29 '12 at 12:32
An irreducible? –  alex.jordan Jul 15 '12 at 0:14

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.

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A (wild) (educated) guess in relation to arithmetic tells me that they could be called irreducible, or even prime, or indecomposable, as Sean Tilson suggested. That is, if we make divisibility more abstract (i.e. require divisibility in a rng, but not thinking about gcd's or lcm's...) –  AdrianM Jan 27 '12 at 17:33
I don't like the term "indecomposable." In a ring with unit, you'd want it to refer to an element which can't be written as the product of two non-unital elements. –  Qiaochu Yuan Jan 27 '12 at 17:46
Perhaps the term simple element could be used. I think it conjures up the appropriate association without it having some other (element-wise) meaning. –  Miha Habič Jan 27 '12 at 18:26
@QiaochuYuan: Isn't that what I have written? The elements of $I$ are non-unital if there is a unit and the composition of $\epsilon$ with the unit map is the identity of $k$. –  Sean Tilson Jan 28 '12 at 3:53
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.