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I am not very familiar with the kind of things people study about rngs which is an acronym people often use for algebraic structures satisfying the ring axioms except for the existence of a multiplicative identity (and whatever is implied by losing this axiom).

I was looking at some old contest problems, mainly from Székely text which features some nice contest problems about rngs but the problems either becomes trivial or meaningless when considered for rings.

I want to invite some discussion about instances where one does research work with rngs and where it is crucial to not assume the existence of a multiplicative identity element and/or where the problem becomes meaningless or trivial if we assume existence of a multiplicative identity.

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    If you take $\mathbb R$ without $1$, then it stops being closed under addition because $1/2+1/2$ is not defined. It stops being a ring, and algebra, a group or even a vector space. – Anixx Aug 13 '24 at 07:14
  • but it fails to be an abelian group in the first place. Am I understanding something wrong? –  Aug 13 '24 at 07:15
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    @Anixx based on the context given in the post it seems OP is asking about if we don't require a multiplicative identity, not that we literally restrict to rings without identity – TY Mathers Aug 13 '24 at 07:20
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    @AlexMathers indeed, I seemingly misunderstood as I though he meant $\mathbb R$, but he means just an arbitrary rng without identity. – Anixx Aug 13 '24 at 07:24

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One place where rng occures naturally is the convolution operator. This operator together with function addition and scalar multiplication forms an associative, distributive and commutative algebra over $\mathbb{R}$. But without identity.

As for pure algebraic properties it is worth noting one important difference between rings and rngs: every ideal in a ring is contained in a maximal ideal. That's not true for rng.

What is true however is that every rng arises as an ideal of some ring. For example due to Dorroh extension.

freakish
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