Problems of rings in abstract algebra

Wernilou

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Mar 25, 2020
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Hey, hello. I need some help with resolving these, teacher said this would help in the test and told us to look around with these exercises. I have been revolving around with these 3, i have the other 3 resolved. These 3 really made my week frustrating. Any help with these would be appreciated please.

1_ If R is a ring and a E R, let AR = {r E R I ar = OR}, Prove that AR is a subring
of R. AR is called the right annihilator of 11.

2_ Let Z* denote the ring of integers with the operations defined as: a+b = a+b-1 and a*b = a+b-a*b, The right operations are the usual addition and multiplication. Prove that Z is isomorphic to Z*

3_ Show that the first ring is not isomorphic to the second. Q and R.

Anything helps please, I know how to know when a set is a ring and more or less homomorphism, but these are really frustrating me. Thank you
 

Jomo

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Sorry but we do not work out problems for students on this site. You should know this by know.

1) Can you at least tell us the definitions of a ring, subring and annihilator? How do you show that a subset of a ring is a subring. You have to know these definitions before you can do these problems.

Lets work with this 1st problem for know. Please post back with the answers to my questions and any work you can do.
 

Wernilou

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Mar 25, 2020
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Sorry but we do not work out problems for students on this site. You should know this by know.

1) Can you at least tell us the definitions of a ring, subring and annihilator? How do you show that a subset of a ring is a subring. You have to know these definitions before you can do these problems.

Lets work with this 1st problem for know. Please post back with the answers to my questions and any work you can do.
A ring is an algebraic system made by a set and two operations, or a group with another operation. A ring needs certain properties to be called a ring. With the first operation we need to make sure it's closed (operated elements of the ring end up in the ring), theres an identity as a+0=a, it's associative with this first operation, it has an inverse with this operation. For the second operation we need it to be closed, associative and needs to be distributive. If it has an identity as a*1=a and also is conmutative (ab=ba) it's called a conmutative ring. It can also have an integral domain when nonzero elements operated with the second operation dont result in 0.

A subring needs to be closed with the first and the second operation, the identity of the ring must be in the subring and every element must have an inverse.

As for the annihilator i dont really know, teacher hasn't reached that yet.

And for the 1 i thought in resolving it as a subring, but the only operation i see is multiplication (ar= 0), so i dont know how to meet the other requirements to know if its a subring.
 

HallsofIvy

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Jan 27, 2012
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In order to show that this is a subring you need to show that it is closed under addition and multiplication, the additive identity is in the subset, that, for any member of the subset, its additive inverse is in the subset, that operations are associative, and that multiplication distributes over addition. Since the operations in the original ring are associative and multiplication distributes over addition and all members of the subset are member of the original ring, it follows immediately that the subset have those properties. If a and b are in the subset, ar= 0 and br= 0 so (a+ b)r= ar+ br= 0+ 0= 0. If a and b are in the subset, ar= 0 and br= 0 so ab(r)= a(br)= a0= 0.
 

pka

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Here is a quick way to show a nonempty subset \(\mathscr{S}\) of a ring \(\mathscr{R}\) is a subring if and only if \(\{a,b\}\subset\mathscr{S}\) implies that \(sb\in\mathscr{S}~\&~a-b\in\mathscr{S}\).
That appears on page 2 of Neal McCoy's The Theory of Rings. That happens to be a most used source.
 

pka

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Jan 29, 2005
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10,453
Here is a quick way to show a nonempty subset \(\mathscr{S}\) of a ring \(\mathscr{R}\) is a subring if and only if \(\{a,b\}\subset\mathscr{S}\) implies that \(sb\in\mathscr{S}~\&~a-b\in\mathscr{S}\).
That appears on page 2 of Neal McCoy's The Theory of Rings. That happens to be a most used source.
EDIT \(ab\in\mathscr{S}~\&~a-b\in\mathscr{S}\).
 
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