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.
A subring is a subset of a given ring that is also a ring under the same operations and with the same identity element.
The intersection of two subrings is also a subring of the original ring. This means that it follows the same operations and identity element as the original ring.
Yes, a ring can have an infinite number of subrings. Any subset of a ring that follows the same operations and identity element can be considered a subring.
Not necessarily. A subring can have the same number of elements as the original ring, but it must still follow the same operations and identity element to be considered a subring.
Subrings allow us to study smaller, more manageable structures within a larger ring. They also help us understand the properties and behaviors of the original ring by looking at the properties and behaviors of its subrings.