Hey there! Sign in to join this conversationNew here? Join for free

Rings or rngs? Watch

Announcements
    • Thread Starter
    Offline

    11
    ReputationRep:
    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?
    Online

    17
    ReputationRep:
    (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...
    Offline

    17
    ReputationRep:
    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.
    Offline

    13
    ReputationRep:
    (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".
    • Thread Starter
    Offline

    11
    ReputationRep:
    (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.
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Break up or unrequited love?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.