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Martin Sleziak
Martin Sleziak
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When you say "unique," how strictly do you mean it? For instance, would you be happy to see examples of algebraic objects that are shown to exist with the axiom of choice, and are unique up to isomorphism?

A possible example that comes to mind is the construction of the injective hull of a module $M$ over a ring. The argument that I know to embed a module into an injective requires Baer's criterionBaer's criterion, which seems to require the axiom of choice. After embedding $M$ into an injective, one must then cut down to a minimal injective submodule containing $M$, and this also seems to use choice.

However, if an injective hull exists then it seems to me that it's unique up to isomorphism without requiring the axiom of choice.

(Notice, this is one case where the object is unique up to isomorphism, but not unique up to unique isomorphism. A similar example is the algebraic closure of a field $K$. I almost quoted this as an example of the phenomenon above, but I realized that the proof of the uniqueness of $\overline{K}$ in Lang's Algebra, for instance, uses choice.)

When you say "unique," how strictly do you mean it? For instance, would you be happy to see examples of algebraic objects that are shown to exist with the axiom of choice, and are unique up to isomorphism?

A possible example that comes to mind is the construction of the injective hull of a module $M$ over a ring. The argument that I know to embed a module into an injective requires Baer's criterion, which seems to require the axiom of choice. After embedding $M$ into an injective, one must then cut down to a minimal injective submodule containing $M$, and this also seems to use choice.

However, if an injective hull exists then it seems to me that it's unique up to isomorphism without requiring the axiom of choice.

(Notice, this is one case where the object is unique up to isomorphism, but not unique up to unique isomorphism. A similar example is the algebraic closure of a field $K$. I almost quoted this as an example of the phenomenon above, but I realized that the proof of the uniqueness of $\overline{K}$ in Lang's Algebra, for instance, uses choice.)

When you say "unique," how strictly do you mean it? For instance, would you be happy to see examples of algebraic objects that are shown to exist with the axiom of choice, and are unique up to isomorphism?

A possible example that comes to mind is the construction of the injective hull of a module $M$ over a ring. The argument that I know to embed a module into an injective requires Baer's criterion, which seems to require the axiom of choice. After embedding $M$ into an injective, one must then cut down to a minimal injective submodule containing $M$, and this also seems to use choice.

However, if an injective hull exists then it seems to me that it's unique up to isomorphism without requiring the axiom of choice.

(Notice, this is one case where the object is unique up to isomorphism, but not unique up to unique isomorphism. A similar example is the algebraic closure of a field $K$. I almost quoted this as an example of the phenomenon above, but I realized that the proof of the uniqueness of $\overline{K}$ in Lang's Algebra, for instance, uses choice.)

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Manny Reyes
Manny Reyes
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When you say "unique," how strictly do you mean it? For instance, would you be happy to see examples of algebraic objects that are shown to exist with the axiom of choice, and are unique up to isomorphism?

A possible example that comes to mind is the construction of the injective hull of a module $M$ over a ring. The argument that I know to embed a module into an injective requires Baer's criterion, which seems to require the axiom of choice. After embedding $M$ into an injective, one must then cut down to a minimal injective submodule containing $M$, and this also seems to use choice.

However, if an injective hull exists then it seems to me that it's unique up to isomorphism without requiring the axiom of choice.

(Notice, this is one case where the object is unique up to isomorphism, but not unique up to unique isomorphism. A similar example is the algebraic closure of a field $K$. I almost quoted this as an example of the phenomenon above, but I realized that the proof of the uniqueness of $\overline{K}$ in Lang's Algebra, for instance, uses choice.)