Abstract Algebra The definition of a Ring

before you learn about rings you should really learn about groups so if you haven't already watched our group video then hit the pause button and check it out I'll wait go on roughly speaking a ring is a set of elements where you can add subtract and multiply but you may not be able to divide and multiplication may not be commutative for example the two by two matrices with real numbers form a ring and matrix multiplication is not commutative by not requiring a multiplication to be commutative rings generalize things like matrices which are pretty important by the way did you notice I said elements instead of numbers hmm that's because some rings consist of numbers while others are made of more exotic objects like polynomials or matrices before we give a more precise definition let's see some examples the integers are a ring you can add subtract and multiply any two integers and you'll get another integer but can you divide two integers hmm well yes but you may not get an integer if you divide 3 by 5 you get a fraction not an integer that's key if you add subtract or multiply two elements in the ring you get another element in the ring we say the ring is closed under addition and multiplication what happens in the ring stays in the ring another example of a ring is the set of polynomials with integer coefficients you can freely add subtract and multiply two such polynomials and you will get a third polynomial of this type but division no I don't think though if you divide X plus 1 by X minus 1 you get X plus 1 over X minus 1 hey it's a thing but it's not a polynomial so now for the official definition a ring is a set of elements with two operations addition and multiplication if you add two elements in a ring you get another element in the ring but earlier I said you can add and subtract and a ring what gives well subtraction is just the same as adding inverses so when I said you can add and subtract I was essentially saying that rings have negative elements and if you add an element to its negative you get zero so in a ring the elements are a group under addition better yet addition is commutative in rings now here's where things get serious if you multiply two elements in a ring you get another element in the ring it's closed under multiplication no surprise there but elements may not have inverses under multiplication we may not be able to divide so while the ring is a nice abelian group under addition things may not be so nice with multiplication but hey at least multiplication is associative and to wrap things up there's a rule which connects addition and multiplication the distributive rule all the numbers you studied an arithmetic obeyed the distributive rule the expressions in algebra do as well so we're good this isn't something new ladies and gentlemen I give you the ring