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This exact game was studied and published in 1970: Henson's "Winning Strategies for the Ideal Game".

In particuar, if $R$ is a single-variable polynomial over any PID then the first player wins. Generally, if $R$ is an integral domain with an element $x$ such that $R/(x)$ is a PID but not a field then the first player wins, and the second player wins the corresponding $R/(x^2)$ game.

This exact game was studied and published in 1970: Henson's "Winning Strategies for the Ideal Game".

In particular, if $R$ is a single-variable polynomial over any PID then the first player wins. Generally, if $R$ is an integral domain with an element $x$ such that $R/(x)$ is a PID but not a field then the first player wins, and the second player wins the corresponding $R/(x^2)$ game.

This exact game was studied and published in 1970: Henson's "Winning Strategies for the Ideal Game".

In particular, if R is a single-variable polynomial over any PID then the first player wins.

This exact game was studied and published in 1970: Henson's "Winning Strategies for the Ideal Game".

In particuar, if $R$ is a single-variable polynomial over any PID then the first player wins. Generally, if $R$ is an integral domain with an element $x$ such that $R/(x)$ is a PID but not a field then the first player wins, and the second player wins the corresponding $R/(x^2)$ game.

This exact game was studied and published in 1970: Henson's "Winning Strategies for the Ideal Game".

In particuar, if R is a single-variable polynomial over any PID then the first player wins.

This exact game was studied and published in 1970: Henson's "Winning Strategies for the Ideal Game".

In particular, if R is a single-variable polynomial over any PID then the first player wins.