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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?

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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.2012-01-28
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    "Rng" is a fancy term for a non-unital ring.2012-01-28
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    When there ever was an abuse of language: Here is one.2012-01-29
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    An irreducible?2012-07-15

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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...)2012-01-27
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    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.2012-01-27
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    Perhaps the term _simple element_ could be used. I think it conjures up the appropriate association without it having some other (element-wise) meaning.2012-01-27
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    @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 $k$.2012-01-28
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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.

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    Wait, how is this set a ring? What is the product of say, a length 10 sequence and a length 20 sequence? I could see concatenation as a product, but then what is their sum? Or do we put an infinite string of 0s at the end of the sequence?2012-07-15
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    @alex.jordan Just imagine everything as an infinituple. I.e. concatenate zeroes after every finite sequence. Then addition and multiplication works the obvious way. And it's a rng, not a ring. Its missing the identity, you see.2012-07-15
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    I see - in time to edit my last comment :) By the way, I purposely did not use the abhorrent term "rng", which just looks like a typo. Someone at some point thought they were being clever - I just think they were being boneheaded. "Nonunital ring" works for me.2012-07-15
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    @alex.jordan Surely you admit that the term "rng" is rather clever. Perhaps I am too easily amused.:)2012-07-15
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    Spoken out loud, it sounds too similar to "ring". Typed, I am left wondering if "ring" was intended, and there is a typo. So it is not good terminology.2012-07-15