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# Rings or rngs? watch

1. This question is aimed at people who have recently learnt/taught algebra: are your rings defined to have a unity element or not?

I ask as I'm sure that I learnt that rings need not have unity, but recent googling suggests that most people think that a ring has unity, and without it's a rng. Is this now standard?
2. (Original post by atsruser)
This question is aimed at people who have recently learnt/taught algebra: are your rings defined to have a unity element or not?

I ask as I'm sure that I learnt that rings need not have unity, but recent googling suggests that most people think that a ring has unity, and without it's a rng. Is this now standard?
Massive discussion on this here: https://en.wikipedia.org/wiki/Talk%3...ty_and_unit.3F

I would hesitate to say there's any consensus; it does seem the tide is flowing towards the "ring has a 1, use rng otherwise" position, but there are clearly enough dissenters that you can't just assume this is the case in any given scenario.

Edit: in fact, the table of 65 books surveyed in the linked page shows that 42 do not define a ring to contain a 1 and only 23 do. (From reading various bits of commentary I think my "tide flowing" comment still holds in that a lot of people currently teaching take the "has a 1" position. But books stay around for a lot of years, so there's a lot of inertia...)

FWIW, I'm sure I was taught rings don't need to contain a 1 as well...
3. I don't think so. I've just looked back over my lecture notes to check and in the algebra course taught to me two years ago (University of Bath) we were taught that a ring need not necessarily have a unity element, but that if it does, it is furthermore a unital ring.

I seem to remember my lecturer bringing up the rng notation as an alternative some people use, but the lecturer didn't use it and we didn't either.

In my view this is one of (many) non-standardised notations in maths. Do you include zero in your Natural Numbers? Does the subset symbol denote a strict or non-strict subset? Different mathematicians use different notation: I see ring/unital ring vs rng/ring as no different to either of those. As long as you make it clear what notation you are using I don't think it matters very much what you choose to use.
4. (Original post by atsruser)
This question is aimed at people who have recently learnt/taught algebra: are your rings defined to have a unity element or not?

I ask as I'm sure that I learnt that rings need not have unity, but recent googling suggests that most people think that a ring has unity, and without it's a rng. Is this now standard?
We were taught (in the late seventies, so apologies for my answer being encrusted with cobwebs) to refer to a ring that had a 1 as a "ring with a 1" or a "ring with unity". Lectures courses tended to go something like "from now on, we shall assume that all rings have a 1 unless otherwise specified".
5. (Original post by Gregorius)
We were taught (in the late seventies, so apologies for my answer being encrusted with cobwebs) to refer to a ring that had a 1 as a "ring with a 1" or a "ring with unity". Lectures courses tended to go something like "from now on, we shall assume that all rings have a 1 unless otherwise specified".
I learnt it at roughly the same time. However, I note that all respondents so far have been from the English university system; maybe there is or has been an Euro-American divergence regarding this definition? I only have two algebra texts both by Americans, and they agree with the defn I am familiar with, however. They're both pretty old though.

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