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nLab > Latest Changes: integral domain

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 27th 2020

    Added a related concept.

    diff, v8, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 27th 2020

    This article claims:

    Another equivalent definition is: an integral domain is any subring of a skewfield. Specifically, any integral domain R is a subring of its field of fractions.

    However, field of fractions claims

    Not every noncommutative integral domain can be embedded at all into a division ring.

    It looks like there is a contradiction between these two claims.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 2nd 2020

    The claim in this article was added in Revision 7 by Toby Bartels on September 15, 2016.

    The claim in field of fractions was added in Revision 1 by Zoran Škoda on July 27, 2011.

    I think Zoran is correct here, though it would be nice to have a specific counterexample.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 2nd 2020

    In the language that I speak, an integral domain is a non-terminal commutative ring with no zero divisors, and my own opinion is that this should be the default. I accept that some mathematicians refer to noncommutative integral domains, but I think they are in the minority.

    According to the Encyclopedia of Mathematics, a counterexample to Toby’s claim can be found in P.M. Cohn’s book “Free rings and their relations”.

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 2nd 2020

    Deleted the incorrect statement.

    diff, v9, current

  6. added link to Wikipedia article

    Anonymous

    diff, v24, current

  7. added fact that every integral domain is a reduced ring.

    Anonymous

    diff, v25, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 24th 2023

    Deleted the claim that “on the nLab” we demand integral domains and commutative, and instead added a remark below the definition on differing conventions.

    diff, v27, current

  9. changed higher algebra - contents to algebra - contents in context sidebar

    Anonymouse

    diff, v30, current

1 to 9 of 9

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