At 3:33, I think you mean "all polynomials with constant coefficient equal to 0" (not those with degree 1 or more). For example, x+1 is a degree 1 polynomial, but is not in x*Z[x].
When did, "There exists a multiplicative identity" stop being a part of the definition of a ring? In my '95 undergrad class it was definitely part of the definition, and a "ring without 1" was called a "rng".
The definition you've seen appears to be a more modern definition. Given that you saw that definition of "ring" and also the use of the word "rng" as early as '95, I suspect you were learning from people who were in the cutting edge of research in areas like universal algebra and category theory (or perhaps people like John Carlos Baez, who claims to have coined the term "rng"). Historically, the definition of ring did not require the existence of a multiplicative identity, but many pure algebraists today feel that this is a mistake, and that morally, a ring should have a multiplicative identity. This hasn't universally caught on, however, and _most_ undergraduate textbooks in algebra _still_ use the older definition. The older definition is also quite useful for some mathematicians in analysis, since some constructions in analysis yield rings without identity.
If rings are not supposed to have an identity, then taking any abelian group A one can define a ring structure by setting xy=0 for every x,y in A. Also, every abelian group M is a module over A.
At 3:33, I think you mean "all polynomials with constant coefficient equal to 0" (not those with degree 1 or more). For example, x+1 is a degree 1 polynomial, but is not in x*Z[x].
In the last example more term will cancel out inside the brackets involving "a".
I really appreciate this video. Thank you
Very good lecture♡
Please make a video for a Differential geometry course.
When did, "There exists a multiplicative identity" stop being a part of the definition of a ring? In my '95 undergrad class it was definitely part of the definition, and a "ring without 1" was called a "rng".
Ring With Unity?
The definition you've seen appears to be a more modern definition. Given that you saw that definition of "ring" and also the use of the word "rng" as early as '95, I suspect you were learning from people who were in the cutting edge of research in areas like universal algebra and category theory (or perhaps people like John Carlos Baez, who claims to have coined the term "rng").
Historically, the definition of ring did not require the existence of a multiplicative identity, but many pure algebraists today feel that this is a mistake, and that morally, a ring should have a multiplicative identity. This hasn't universally caught on, however, and _most_ undergraduate textbooks in algebra _still_ use the older definition. The older definition is also quite useful for some mathematicians in analysis, since some constructions in analysis yield rings without identity.
If rings are not supposed to have an identity, then taking any abelian group A one can define a ring structure by setting xy=0 for every x,y in A. Also, every abelian group M is a module over A.
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Instead of a having an inverse, how about if it is "only" and adjoint relation. Like a residuated lattice.
hello, what do you mean by 3 times 5 =15 =0 inside z15
Like how with a clock (Z12), 5+8=13mod12=1
Z15 means whole numbers mod 15.
Mod (or the modulus operator) refers to the remainder when dividing.
2:44 functions R -> R are a ring under addition and COMPOSITION (he doesn't mean addition and multiplication)
but multiplication in abstract algebra does not refer to "normal" multiplication, like in R or Z.
Is the addition just pointwise addition? If so wouldn’t pointwise multiplication also form a ring?
The set of functions R->R is not a ring under addition and composition. In general, the distributive property does not hold.