Talk:Semifield

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In the text, there is a statement

and (S,·) is a division ring that is not assumed to be commutative or associative

whereas, according to an article about rings, a ring has to be associative. How does this fit together? Additionally, I don't think (S,·) can be a ring at all, because a ring rquires two binary operations.--Slow Phil (talk) 17:13, 4 June 2012 (UTC)[reply]

If the term "ring" is used without any qualifiers, then it is associative, however, especially in universal algebra, non-associative rings are also considered. In the opposite direction, in commutative algebra, "ring" typically means "a commutative associative ring with a unit". I don't really understand Q2: semifield by either definition given in the article possesses two binary operations, "addition" and "multiplication". Arcfrk (talk) 22:58, 4 June 2012 (UTC)[reply]

My concern is this passage: "A variation of this definition arises if S contains an absorbing zero that is different from the multiplicative unit e, it is required that the non-zero elements be invertible, and a·0 = 0·a = 0." AFAIK, this "variation" is the most common paradigm, and semirings without neutral "additive" element are the exception. At least, one of the given citations, Golan's book, defines semirings that way. I don't have access to the other given resource, but I know a third resource, Gondran and Minoux, concurs with Golan on including a neutral additive element by default. I'm of the opinion the article should reflect that. Other input? Rschwieb (talk) 14:31, 6 December 2018 (UTC)[reply]