The discussion revolves around the properties of subrings within the context of ring theory, specifically focusing on the intersection of two subrings and whether this intersection itself forms a subring.
There is ongoing exploration of the necessary conditions for the intersection to be a subring, with some participants suggesting that the original argument lacks clarity. Others provide insights into the definitions and properties of rings and subrings, indicating a productive dialogue without a clear consensus.
Participants note the importance of including zero in any ring and discuss the multiplicative identity's role in subrings, highlighting differing assumptions about definitions in ring theory.
How does this follow? the way you have said it, c is just a member of C and has nothing to do with a and b. (There may be many different a and b such that a+ b= c for a given c.)pivoxa15 said:Homework Statement
Show that the intersection of any two subrings of a ring is a subring.
The Attempt at a Solution
It seems abstract.
suppose a+b=c and a*b=d
Then if c is in A and B (where A and B are subrings) then the intersection of A and B denoted by C contains c and [/b]if C contains more the one element then it must contain a and b.[/b]
I think the definition that rings have '1' is sufficiently pervasive that it should be assumed when not otherwise specified... honestly, the magma algebra system is the only context I've ever seen where a "ring" is not used to mean a unital associative algebra.Mystic998 said:0 has to be in any ring, but a subring doesn't doesn't have to contain the multiplicative identity (assuming the original ring even has one). A good example is 2Z in Z.
Of course, that's entirely beside the point, but the question has already been answered, and I like being persnickety.
HallsofIvy said:How does this follow? the way you have said it, c is just a member of C and has nothing to do with a and b. (There may be many different a and b such that a+ b= c for a given c.)
You need to show "if a and b are in C, then -a is in C, a+ b is in C, and a*b is in C. Since 0 and 1 must be in any ring, they must be in A and B and so in C.