Trying to understand more about polynomials in noncommuting variables.
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I'm working on a project and I will need to look at some theory regarding polynomials in noncommuting variables. I have a 1st ed. copy of Rotman's Advanced Modern Algebra, which I've been getting some of my theory from (There is a pdf version of the 2nd ed. you can find online by googling). On page 724 of the same text is where he offers the definition for the Ring of polynomials over $R$ in noncommuting variables $X$, denoted $Rlangle Xrangle$. The definition verbatim is as follows
$textbf{Definition}$: If $R$ is a commutative ring and $V$ is a free $R$-module with basis $X$, then $T(V)$ is called the ring of polynomials over $R$ in noncommuting variables $X$, and it is denoted by $Rlangle Xrangle$.
On page 725 he continues on just a little about the subject but its not much help to my own understanding. So what I was hoping to ask for were some recommendations of resources regarding this topic that might aid in my understanding of the material. I've tried to look for some results myself but I haven't found anything helpful yet; I have a feeling that it might be due to the fact that I'm not familiar enough with the algebra jargon currently in use.
polynomials ring-theory modules free-modules graded-algebras
asked Nov 16 at 1:38
Thomas Lawrence
1
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I'm working on a project and I will need to look at some theory regarding polynomials in noncommuting variables. I have a 1st ed. copy of Rotman's Advanced Modern Algebra, which I've been getting some of my theory from (There is a pdf version of the 2nd ed. you can find online by googling). On page 724 of the same text is where he offers the definition for the Ring of polynomials over $R$ in noncommuting variables $X$, denoted $Rlangle Xrangle$. The definition verbatim is as follows
$textbf{Definition}$: If $R$ is a commutative ring and $V$ is a free $R$-module with basis $X$, then $T(V)$ is called the ring of polynomials over $R$ in noncommuting variables $X$, and it is denoted by $Rlangle Xrangle$.
On page 725 he continues on just a little about the subject but its not much help to my own understanding. So what I was hoping to ask for were some recommendations of resources regarding this topic that might aid in my understanding of the material. I've tried to look for some results myself but I haven't found anything helpful yet; I have a feeling that it might be due to the fact that I'm not familiar enough with the algebra jargon currently in use.
polynomials ring-theory modules free-modules graded-algebras
asked Nov 16 at 1:38
Thomas Lawrence
1
1
What do you want to know about it?
– Qiaochu Yuan
Nov 16 at 4:11
In particular, I would just like clarity in my own understanding of the definition (maybe through illuminating examples) and the beginings of any kind application of the subject.
– Thomas Lawrence
Nov 16 at 6:48
In all frankness, anything. Just more than what I have in front of me. @QiaochuYuan
– Thomas Lawrence
Nov 16 at 6:58
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm working on a project and I will need to look at some theory regarding polynomials in noncommuting variables. I have a 1st ed. copy of Rotman's Advanced Modern Algebra, which I've been getting some of my theory from (There is a pdf version of the 2nd ed. you can find online by googling). On page 724 of the same text is where he offers the definition for the Ring of polynomials over $R$ in noncommuting variables $X$, denoted $Rlangle Xrangle$. The definition verbatim is as follows
$textbf{Definition}$: If $R$ is a commutative ring and $V$ is a free $R$-module with basis $X$, then $T(V)$ is called the ring of polynomials over $R$ in noncommuting variables $X$, and it is denoted by $Rlangle Xrangle$.
On page 725 he continues on just a little about the subject but its not much help to my own understanding. So what I was hoping to ask for were some recommendations of resources regarding this topic that might aid in my understanding of the material. I've tried to look for some results myself but I haven't found anything helpful yet; I have a feeling that it might be due to the fact that I'm not familiar enough with the algebra jargon currently in use.
polynomials ring-theory modules free-modules graded-algebras
asked Nov 16 at 1:38
Thomas Lawrence
1
I'm working on a project and I will need to look at some theory regarding polynomials in noncommuting variables. I have a 1st ed. copy of Rotman's Advanced Modern Algebra, which I've been getting some of my theory from (There is a pdf version of the 2nd ed. you can find online by googling). On page 724 of the same text is where he offers the definition for the Ring of polynomials over $R$ in noncommuting variables $X$, denoted $Rlangle Xrangle$. The definition verbatim is as follows
$textbf{Definition}$: If $R$ is a commutative ring and $V$ is a free $R$-module with basis $X$, then $T(V)$ is called the ring of polynomials over $R$ in noncommuting variables $X$, and it is denoted by $Rlangle Xrangle$.
On page 725 he continues on just a little about the subject but its not much help to my own understanding. So what I was hoping to ask for were some recommendations of resources regarding this topic that might aid in my understanding of the material. I've tried to look for some results myself but I haven't found anything helpful yet; I have a feeling that it might be due to the fact that I'm not familiar enough with the algebra jargon currently in use.
polynomials ring-theory modules free-modules graded-algebras
polynomials ring-theory modules free-modules graded-algebras
asked Nov 16 at 1:38
Thomas Lawrence
1
asked Nov 16 at 1:38
Thomas Lawrence
1
asked Nov 16 at 1:38
Thomas Lawrence
1
asked Nov 16 at 1:38
Thomas Lawrence
1
asked Nov 16 at 1:38
Thomas Lawrence
1
1
1
What do you want to know about it?
– Qiaochu Yuan
Nov 16 at 4:11
In particular, I would just like clarity in my own understanding of the definition (maybe through illuminating examples) and the beginings of any kind application of the subject.
– Thomas Lawrence
Nov 16 at 6:48
In all frankness, anything. Just more than what I have in front of me. @QiaochuYuan
– Thomas Lawrence
Nov 16 at 6:58
add a comment |
1
What do you want to know about it?
– Qiaochu Yuan
Nov 16 at 4:11
In particular, I would just like clarity in my own understanding of the definition (maybe through illuminating examples) and the beginings of any kind application of the subject.
– Thomas Lawrence
Nov 16 at 6:48
In all frankness, anything. Just more than what I have in front of me. @QiaochuYuan
– Thomas Lawrence
Nov 16 at 6:58
1
1
What do you want to know about it?
– Qiaochu Yuan
Nov 16 at 4:11
What do you want to know about it?
– Qiaochu Yuan
Nov 16 at 4:11
In particular, I would just like clarity in my own understanding of the definition (maybe through illuminating examples) and the beginings of any kind application of the subject.
– Thomas Lawrence
Nov 16 at 6:48
In particular, I would just like clarity in my own understanding of the definition (maybe through illuminating examples) and the beginings of any kind application of the subject.
– Thomas Lawrence
Nov 16 at 6:48
In all frankness, anything. Just more than what I have in front of me. @QiaochuYuan
– Thomas Lawrence
Nov 16 at 6:58
In all frankness, anything. Just more than what I have in front of me. @QiaochuYuan
– Thomas Lawrence
Nov 16 at 6:58
add a comment |
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What do you want to know about it?
– Qiaochu Yuan
Nov 16 at 4:11
In particular, I would just like clarity in my own understanding of the definition (maybe through illuminating examples) and the beginings of any kind application of the subject.
– Thomas Lawrence
Nov 16 at 6:48
In all frankness, anything. Just more than what I have in front of me. @QiaochuYuan
– Thomas Lawrence
Nov 16 at 6:58