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Yesterday we cover the definition of rings briefly in class. My teacher wrote that an algebraic structure is called ring $(R,+,.)$ if the set R is abelian group under addition, and multiplication is associative over addition etc.Here, what I stuck in is $(R,+,.)$, all books use addition and multiplication with the definition of R must be group under addition.

My question is that should we always use addition and multiplication as binary operations while defining the rings. Moreover, is the order of binary operations important. For example, if it were $(R,.,+)$ instead of $(R,+,.)$, then would we still say that it must be group under addition ? Otherwise, would we say that it must be group under multiplication ? Briefly can we say that if it is defined as $(S,\alpha,\beta)$, then it can be a ring if the set $S$ is an abelian group under operation of $\alpha$ no matter what $\alpha$ is, and it satify other conditions under $\beta$ no matter what operation the $\beta$ is. So, the order of binary operations determine the role of opeartions in definition. Can you explain or correct me ?

  • It rather looks like you have qualms about notation rather than definition. There really isn't any universal truth to why we use the notation we do. We try to pick it to be predictable, convenient, and memorable as possible so that it is easy to work with. The symbols don't determine the meaning, we choose the symbols to remind us of the meaning. – rschwieb Dec 21 '23 at 12:33
  • @rschwieb what do you think about my thought about the notation. Can you please look at my comment under the accepted answer –  Dec 21 '23 at 13:19
  • hımm, so the order of operations determine their roles. Yes and no. "Yes" because that's the convention most of us use and "no" because it could just as well have been a convention to do it the other way around. The order doesn't determine their roles, our convention to write it in that order is what alerts us to which is which. I think mainly this is an outgrowth of considering $(R,+)$ first as an abelian group, then layering on another operation after the fact to extend it to $(R,+,\cdot)$. The second operation is "less nice" and "later" somehow. – rschwieb Dec 21 '23 at 16:54
  • @rschwieb look at the comments between me and Eric, I think that you miss the thought of ring is an ordered triple – Not a Salmon Fish Dec 21 '23 at 17:48
  • @rschwieb look at here – Not a Salmon Fish Dec 21 '23 at 18:18
  • Such (syntactic) matters are considered more precisely in model theory, e.g. see Hodges' remarks here. – Bill Dubuque Dec 21 '23 at 18:50
  • @BillDubuque chatgpt says: we can use different symbols to denote the operations in a ring, as long as the operations satisfy the axioms of a ring. The symbols of addition and multiplication are commonly used, but they are not mandatory. For example, in a Boolean ring, the addition operation is defined as the Boolean operation of OR, and the multiplication operation is defined as the Boolean operation of AND. In this case, we use the symbols "+" and "·" to denote the operations, respectively. – Not a Salmon Fish Dec 21 '23 at 19:21
  • @BillDubuque the order of binary symbols in ring notation is important. The symbols used for addition and multiplication in a ring are typically denoted as "+" and "⋅" respectively. However, the order in which these symbols are written can affect the interpretation of the expression. The most common way to write the expression is to use the infix notation, where the operator is placed between the operands. For example, if a and b are elements of a ring R, we write a + b and a ⋅ b to denote the addition and multiplication operations, respectively. – Not a Salmon Fish Dec 21 '23 at 19:22
  • In terms of the axioms of a ring, the "+" symbol is typically used to denote the operation that satisfies the axioms of a commutative group, which include closure, associativity, the existence of an identity element, and the existence of inverse elements. – Not a Salmon Fish Dec 21 '23 at 19:22
  • @BillDubuque I agree with OP, what do you think ? – Not a Salmon Fish Dec 21 '23 at 19:23
  • @Not As is usual with overloaded math notation what matters is that it is uniquely readable in the ambient context. In many contexts it is unambiguous which operations are addition and multiplication so no further conventions are needed to distinguish them. Otoh if the notation is to be parsed by a machine (e.g. a computer algebra system) then more pecise notation may be necessary. – Bill Dubuque Dec 21 '23 at 19:32
  • Related: some algebraic structures (e.g. groups) have a variety of common axiomatizations using various operations. But the choice of which operations to use in the definition may be constrained by the context, e.g. for groups we need to include the inverse operation in the signature in order for the universal notions of homomorphism and substructure to specialize to the classical notions for groups, cf. here. – Bill Dubuque Dec 21 '23 at 20:14

2 Answers2

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My question is that should we always use addition and multiplication as binary operations while defining the rings.

Do you mean the symbols $+$ and $\cdot$? You don't have to, but by convention one of these (under which $R$ is an abelian group under addition) is called addition, and the other is called multiplication, so these are the most natural choices of symbol to use.

(This is especially true since rings are meant to be "integer-like", and the integers naturally have notions of addition and multiplication. You could even argue that this is reflected in the name of the structure, in a sense, though this is tenuous. Some worthwhile reading on the etymology here on MO.)

Of course, in other contexts, especially with particular examples, others could be used.

For example, if it were $(R,.,+)$ instead of $(R,+,.)$, then would we still say that it must be group under addition ?

This is another matter of convention. If you are consistent in whatever framework you're working in (e.g. homework you're writing, textbook you're writing, paper you're writing), and your definitions clear, I don't think anyone is going to care about the order, so long as you still maintain the obvious that

  • $+$ is addition
  • $\cdot$ is multiplication
  • $(R,+)$ is an abelian group
  • $(R,\cdot)$ satisfies its list of axioms
  • distributivity is satisfied

This doesn't address the implicit question of "okay, so why do we list addition first in the tuple?" to which I have no good answer.

On the other hand, if your question is "can I notate addition by $\cdot$ and multiplication by $+$?", you can, but there is the natural follow-up question of "why in the world would you use such a counterintuitive notation for a generic ring?" Give the operations whatever symbols you please and the technicalities will work out, but the choice of notation should be deliberate and made for a reason.

PrincessEev
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  • actually this doesnt fully answer my question. I am looking for the definiton of ring without depending on addtion and multiplication. what do you think about my alpha beta definition ? I am trying to understand that whether the order of operation in parentheses determine their role. Or , the order of opeartions doesnt affect their roles in definiton of ring but their functions determine their roles. –  Dec 21 '23 at 07:21
  • The symbols chosen to denote the operations, and the order of the symbols in the notation for a ring is totally irrelevant to the underlying definition. The symbols are a convention established so you don't have to say the entire definition over again when writing a new ring. Most people use "+" and "$\cdot$" and put them in the order $(R,+,\cdot)$ because it uses familiar symbols and is memorable once explained. For sure you could obscure things and use $\alpha$ for addition and $\beta$ for multiplication and scramble the symbols into $(\beta, R, \alpha)$... at the cost of causing confusion. – rschwieb Dec 21 '23 at 12:31
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You can say that $(R,\alpha,\beta)$ is a ring for any binary operations $\alpha$ and $\beta$ such that $(R,\alpha)$ is an abelian group and [blah blah blah whatever else]. For the sake of learning I'd say pretend that the order matters so the second element $\alpha$ of the tuple will be the one for which $(R,\alpha)$ is a group. But if you walk into a seminar one day and say $(R,\cdot,+)$ is a ring nobody is going to be mad or confused as long as you really mean $(R,+)$ is the abelian group (you would never even write the full triple in this circumstance anyways).

Alex Mathers
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  • hımm, so the order of operations determine their roles. The first operation from the left side is used to check whether the set is group and the last operation determine whether the set satisfy the other conditions. IN $(R,\alpha,\beta)$ , we check whether it is abelian group with respect to $\alpha$, and in $(R,\beta,\alpha)$ we check whether it is abelian group with respect to $\beta$,. Am I right ? –  Dec 21 '23 at 07:25
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    @junior_student Yes you can take this as correct. The truth is any rings people actually talk about in practice have a clear addition operation, a clear multiplication operation, and we don't bother to write the whole triple $(R,+,\cdot)$; rather we just say "let $R$ be a ring". So people don't really actually worry about this kind of thing in practice, there are never situations where it's ambiguous which operation is the "addition" and which is the "multiplication" – Alex Mathers Dec 21 '23 at 07:27