elias001
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- TL;DR Summary
- I want to know if my attempt at translating definition of minimal ideal in symbolic logic notation, into both negation and non negation form are correct.
The Definitions below are taken from the following books:
Fundamentals of abstract algebra by: Malik, Morderson, Sen
Rings, Modules and Algebras by: Adamson
First Course in Module Theory by: Keating
Basic Abstract Algebra by: Bhattacharya, Jain Nagpaul
How to Prove it by Dan Velleman
Definition 1: A left ideal ##M## of ##S## is minimal if every left ideal ##I## of ##S## included in ##M## coincides with ##M##; similarly for right ideals.
[From Adamson]
Definition 2: Let ##R## be a ring, and let ##E_{\ast}## be the set of non-zero left ideals in ##R##. Then the inclusion relation is a relation of order in ##E_{\ast}##; if ##L## is a minimal element of ##E_{\ast}## under this relation it is called a minimal left ideal of of ##R##.
[From: Bhattacharya, Jain Nagpaul]
Definition 3: A ring (left) ideal ##I## in a ring ##$R## is called minimal if ##(i)## ##I\neq \{0\}##, and ##(ii)## if ##J## is a nonzero right (left) ideal of ##R## contained in ##I##, then ##J=I##.
Definitions in terms of negation
[From: Keating, M.E's First Course in Module Theory]
Definition 4: A left ideal ##I## of a ring ##R## is minimal if ##I## is nonzero and there is no left ideal ##J## with $$0\subset J\subset I.$$
[From: Malik, Morderson, Sen]
Definition 5: An ideal ##I## of a ring ##R## is called a minimal ideal if ##I\neq\{0\}## and there does not exist any ideal ##J## of ##R## such that ##\{0\}\neq J\subset I##.
[From: Velleman]
Definition 6: Suppose ##R## is a partial order on a set ##A,B\subset A##, and ##b\in B##. Then ##b## is called an ##R-##smallest element of ##B## if ##\forall x\in B(bRx)##. It is called an ##R-##minimal element if ##\sim\exists x\in B(xRb\wedge x\neq b)##.
Note: ##\sim\exists x\in B(xRb\wedge x\neq b)##. is equivalent to ##\forall x\in B(xRb\to x=b)##.
I am trying to write out in terms of symbolic logic notation the definition of minimal ideal for commutative rings. The five different definitions above are the ones I found in different texts and an online note. Two of them are written in terms of negations. The last definition is a definition for minimal elements for partial orders. I understand that minimal ideal is a minimal elements for ideals in the partial order of inclusion. The attempts I have tried, I am not sure if it reflects what it says in the various equivalent definitions.
Let ##R## be a commutative ring, let the notation ##I\triangleleft R## denote ##I## being an ideal of ##R##, then
In terms of non negation
##\Bigl((I\triangleleft R)\wedge (J\triangleleft R) \wedge(I\neq R)\wedge(J\neq R)\Bigr)\wedge\Bigl( \forall J\bigl( (J\subseteq I)\wedge J\neq\{0\})\to (J=I))\Bigr)\quad (1)##.
In terms of negation
##\Bigl((I\triangleleft R)\wedge (J\triangleleft R) \wedge(I\neq R)\wedge(J\neq R)\Bigr)\wedge \Bigl( \neg\exists J\bigl((J\subseteq I)\wedge J\neq\{0\})\wedge (J\neq I)\Bigr) \quad (2)##.
Fundamentals of abstract algebra by: Malik, Morderson, Sen
Rings, Modules and Algebras by: Adamson
First Course in Module Theory by: Keating
Basic Abstract Algebra by: Bhattacharya, Jain Nagpaul
How to Prove it by Dan Velleman
Definition 1: A left ideal ##M## of ##S## is minimal if every left ideal ##I## of ##S## included in ##M## coincides with ##M##; similarly for right ideals.
[From Adamson]
Definition 2: Let ##R## be a ring, and let ##E_{\ast}## be the set of non-zero left ideals in ##R##. Then the inclusion relation is a relation of order in ##E_{\ast}##; if ##L## is a minimal element of ##E_{\ast}## under this relation it is called a minimal left ideal of of ##R##.
[From: Bhattacharya, Jain Nagpaul]
Definition 3: A ring (left) ideal ##I## in a ring ##$R## is called minimal if ##(i)## ##I\neq \{0\}##, and ##(ii)## if ##J## is a nonzero right (left) ideal of ##R## contained in ##I##, then ##J=I##.
Definitions in terms of negation
[From: Keating, M.E's First Course in Module Theory]
Definition 4: A left ideal ##I## of a ring ##R## is minimal if ##I## is nonzero and there is no left ideal ##J## with $$0\subset J\subset I.$$
[From: Malik, Morderson, Sen]
Definition 5: An ideal ##I## of a ring ##R## is called a minimal ideal if ##I\neq\{0\}## and there does not exist any ideal ##J## of ##R## such that ##\{0\}\neq J\subset I##.
[From: Velleman]
Definition 6: Suppose ##R## is a partial order on a set ##A,B\subset A##, and ##b\in B##. Then ##b## is called an ##R-##smallest element of ##B## if ##\forall x\in B(bRx)##. It is called an ##R-##minimal element if ##\sim\exists x\in B(xRb\wedge x\neq b)##.
Note: ##\sim\exists x\in B(xRb\wedge x\neq b)##. is equivalent to ##\forall x\in B(xRb\to x=b)##.
I am trying to write out in terms of symbolic logic notation the definition of minimal ideal for commutative rings. The five different definitions above are the ones I found in different texts and an online note. Two of them are written in terms of negations. The last definition is a definition for minimal elements for partial orders. I understand that minimal ideal is a minimal elements for ideals in the partial order of inclusion. The attempts I have tried, I am not sure if it reflects what it says in the various equivalent definitions.
Let ##R## be a commutative ring, let the notation ##I\triangleleft R## denote ##I## being an ideal of ##R##, then
In terms of non negation
##\Bigl((I\triangleleft R)\wedge (J\triangleleft R) \wedge(I\neq R)\wedge(J\neq R)\Bigr)\wedge\Bigl( \forall J\bigl( (J\subseteq I)\wedge J\neq\{0\})\to (J=I))\Bigr)\quad (1)##.
In terms of negation
##\Bigl((I\triangleleft R)\wedge (J\triangleleft R) \wedge(I\neq R)\wedge(J\neq R)\Bigr)\wedge \Bigl( \neg\exists J\bigl((J\subseteq I)\wedge J\neq\{0\})\wedge (J\neq I)\Bigr) \quad (2)##.