How to prove the intersection of two subrings is a subring?
The intersection of two subrings of a given ring is a fundamental concept in ring theory, which is a branch of abstract algebra. Let's break down the key terms and concepts involved:
Ring: A ring is an algebraic structure consisting of a set equipped with two binary operations, usually denoted as addition (+) and multiplication (·), that satisfy the following properties for all elements a, b, and c in the set: a. Closure under addition and multiplication: a + b and a · b are both elements of the ring. b. Associativity: (a + b) + c = a + (b + c) and (a · b) · c = a · (b · c). c. Existence of an additive identity (0): There exists an element 0 such that a + 0 = a for all a in the ring. d. Existence of additive inverses: For every element a, there exists an element -a such that a + (-a) = 0. e. Commutativity of addition (optional): a + b = b + a for all a and b in the ring. f. Distributive property: a · (b + c) = (a · b) + (a · c) and (b + c) · a = (b · a) + (c · a) for all a, b, and c in the ring.
Subring: A subring is a subset of a given ring that is itself a ring with respect to the same addition and multiplication operations defined on the original ring. In other words, it's a subset that forms a ring under the same operations.
Now, let's discuss the intersection of two subrings:
Given two subrings A and B of a ring R, the intersection of A and B is denoted as A ∩ B. It is defined as the set of elements that belong to both A and B.
In other words: AS∩B={x∈R∣x∈Aand x∈B}.A∩B={x∈R∣x∈A and x∈B}.
Properties of the intersection of subrings:
Closure: The intersection A ∩ B is also a subring of R. This means that it is closed under both addition and multiplication.
Identity element: If both A and B have the same multiplicative identity element (1), then A ∩ B will also have the same multiplicative identity.
Inverses: If an element x has an additive inverse in both A and B, then it will have an additive inverse in A ∩ B.
Distributive property: A ∩ B will satisfy the distributive property with respect to the operations of the original ring R because A and B are subrings that already satisfy these properties.
In summary, the intersection of two subrings of a ring is itself a subring that inherits the algebraic properties of the original ring, such as closure, identity elements, inverses, and the distributive property.
Let S1 and S2 be two subrings of a ring R. Then S1∩S2 is not empty since at least 0⋿ S1∩S2.
Now in order to prove that S1∩S2 is subring, it is sufficient to prove that
1. a⋿ S1∩S2,b⋿ S1∩S2
a-b⋿ S1∩S2
2. a⋿ S1∩S2,b⋿ S1∩S2
ab⋿ S1∩S2
We have
a⋿ S1∩S2
a⋿ S1,a⋿ S2
b⋿ S1∩S2
b⋿ S1,b⋿ S2
Now S1 and S2 are both subrings.
such that
a⋿ S1,b⋿ S1
a-b⋿ S1 and ab⋿ S1
and a⋿ S2,b⋿ S2
ab⋿ S2.
Now a-b ⋿ S1,a-b⋿ S2
a-b⋿ S1∩S2
ab⋿ S1,ab⋿ S2
ab⋿ S1∩S2.
Thus
a⋿ S1∩S2,
b⋿ S1∩S2
a-b⋿ S1∩S2.and
ab⋿ S1∩S2.
so that
S1∩S2 is a subring of R.
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