Ring Theory: Proving Subrings and Ring Generation

In summary, the first question asks to show that the intersection of all subrings containing a subset A is also a subring of the original ring R. The second question asks to prove that in a ring where 1 is not equal to 0, the sets ∅, {0} and {1} all generate the same ring.
  • #1
Poirot1
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Two questions

(1)For R a ring and A a subset of R, let s(A) denote the set of all subrings of R that contain A (including R itself). Show that the intersection of all these subrings is itself a subring of R.

(2)Suppose that 1 is not equal to 0 in R. Show that the sets ∅, {0} and {1} all generate the same ring in R.​


Thanks
 
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  • #2
Poirot said:
Two questions

(1)For R a ring and A a subset of R, let s(A) denote the set of all subrings of R that contain A (including R itself). Show that the intersection of all these subrings is itself a subring of R.

(2)Suppose that 1 is not equal to 0 in R. Show that the sets ∅, {0} and {1} all generate the same ring in R.​


Thanks
1) Check whether the following conditions are met:
a) 0 is in s(A)
b) a - b is in s(A) whenever a and b are in s(A)
c) ab is in s(A) whenever a and b is are in s(A)
 

FAQ: Ring Theory: Proving Subrings and Ring Generation

What is ring theory?

Ring theory is a branch of abstract algebra that deals with the properties and structure of rings, which are algebraic structures that consist of a set of elements with operations of addition and multiplication defined on them.

What is a subring?

A subring is a subset of a ring that contains the identity element and is closed under the ring operations of addition, multiplication, and additive inverse. This means that a subring is itself a ring with the same operations and identity element as the original ring.

How do you prove that a subset is a subring?

In order to prove that a subset is a subring, you need to show that it satisfies the definition of a subring. This includes demonstrating that the subset is closed under addition, multiplication, and additive inverse, and that it contains the identity element of the original ring.

What is meant by "ring generation"?

Ring generation is the process of creating a ring by starting with a set of elements and then using the ring operations of addition and multiplication to generate all possible combinations of those elements. This can result in a larger ring with more elements than the original set.

Can a subring be generated by a smaller set of elements?

Yes, a subring can be generated by a smaller set of elements. This is because a subring only needs to satisfy the properties of a ring, which may not require all possible combinations of the original elements. Additionally, some elements in the original set may be redundant and can be removed without affecting the subring.

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