Definition of minimal ideal using symbolic logic notation

[elias001]

TL;DR

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

$\Bigl((I\triangleleft R)\wedge (J\triangleleft R) \wedge(I\neq R)\wedge(J\neq R)\Bigr)\wedge \Bigl( \neg\exists J\bigl((J\subseteq I)\wedge J\neq\{0\})\wedge (J\neq I)\Bigr) \quad (2)$.

 

[fresh_42]

Definitions one to five are ok, but it should be noted that definition one allows zero ideals to be called minimal, and the others don't. A matter of taste, but I think Adamson has simply forgotten to rule out zero ideals since $\{0\}$ is automatically minimal and you don't want to bother with this trivial case every time you mention a minimal idea. So let's pretend that the zero ideal is not a minimal ideal.

Definition six is a bit clumsy and, in my mind, unnecessarily complicated. I further assume that $\sim$ should mean not, which is usually written as $\lnot$ since $\sim$ is reserved for equivalence relations.

The main difference between $R$-smallest and $R$-minimal is due to the partial order we have, i.e., no total order. That means we can have different chains of, say, inclusions. For example, we have
$$
\ldots \subseteq 2^n\mathbb{Z} \subseteq 2^{n-1}\mathbb{Z}\subseteq \ldots\subseteq 8\mathbb{Z}\subseteq 4\mathbb{Z}\subseteq 2\mathbb{Z}\subseteq \mathbb{Z}
$$
and
$$
\ldots \subseteq 3^n\mathbb{Z} \subseteq 3^{n-1}\mathbb{Z}\subseteq \ldots\subseteq 27\mathbb{Z}\subseteq 9\mathbb{Z}\subseteq 3\mathbb{Z}\subseteq \mathbb{Z}
$$
which are different chains of inclusions, and in this case, without a minimal element.

An example for a ring with minimal elements is $\mathbb{F}[x]/\bigl\langle x^n \bigr\rangle ,$ but we can still have different chains of inclusions that have no ideals in common. Now, an $R$-smallest element is included in every such a chain, for example the zero ideal $b=\{0\}$ in case $\{0\}\in B,$ and the $R$-minimal elements are the left-most ideals of all such chains in $B.$

I would forget about definition six as long as you aren't particularly studying orders in general. Important is only the fact that inclusion is only a partial order and not a total order, as my example of ideals chains in $\mathbb{Z}$ shows. You cannot automatically say which is larger than the other if they aren't included in each other.

Now to your logical expressions. First of all, why did you exclude $I=R$? I don't see this case excluded in any of the definitions above. $\mathbb{Q}$ is the minimal ideal of $\mathbb{Q}.$ I would drop the entire expressions on the left before the $\wedge$ sign and replace it with "$\{0\}\neq I \trianglelefteq R$ is minimal iff".

If you want to formalize that $I,J$ are ideals, you could either include this information in the second expressions like
$$
I\neq \{0\}\wedge I\trianglelefteq R \wedge \left((\forall \,J\trianglelefteq R\, : \, J\neq \{0\} \wedge J\subseteq I) \longrightarrow J=I\right)
$$
or make being an ideal different from zero a logical predicate, e.g.
$$
J\,P(R) \longleftrightarrow J\in P(R) \longleftrightarrow J\neq \{0\} \wedge J\trianglelefteq R
$$
and write that $I\, P(R)$ is minimal iff
$$
\forall\,J\,P(R) \left(J\subseteq I\, P(R)\longrightarrow J=I\right)
$$
but I have to admit that I'm not a logician and you should take this with caution. The negation looks fine to me.

 

[elias001]

@fresh_42 if instead of minimal ideal, but changed to minimal prime ideal. I can say something like "let $P$ denote the prime ideal...." Meaning, I don't have to write out fully in symbolic logic notation of a prime ideal and replace everywhere the occurrence of the instance for ideal with minimal ideal $P$ in the translated expression for minimal ideal?

 

[fresh_42]

@fresh_42 if instead of minimal ideal, but changed to minimal prime ideal. I can say something like "let $P$ denote the prime ideal...." Meaning, I don't have to write out fully in symbolic logic notation of a prime ideal and replace everywhere the occurrence of the instance for ideal with minimal ideal $P$ in the translated expression for minimal ideal?

I think I didn't understand you.

Do you want to use logical symbolism because you study logic, or because you want to clarify the problem statements?

In the first case, I would strongly recommend following the textbook you have in hand. And if there is more than one, close everyone but one. Different books usually have different notations. Avoid confusing them.

In the second case, I would try to make notations as simple as possible. Define sets of ideals (of any kind you need) and write down the statements by using these sets, either with the existence of an element of these sets, possibly with additional constraints (i.e. elements of a non-empty subset), or with all quantifiers for all elements of the set. It is confusing if you try to combine the definition of these sets with the logical statements themselves. It distracts from the purpose namely to prove the statements.