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Krypton is a noble gas and has a complete valence shell. For it to be ionized, or to achieve ionization, it would take an insane amount of energy to remove an electron from a complete shell. It is easier for Mg to remove an electron than Krypton.
Yes, Kr is stable since it has a filled valence shell, so it holds onto its electrons much more strongly than Mg does. Even though Mg is smaller, it only has 2 valence electrons, and prefers to lose those valence electrons and form a cation that resembles the fully stable octet of neon. Since it's harder to remove the valence electrons of Kr, it has a greater first IE.
As a general rule, elements with a full outer valence shell want to stay that way. Taking away an electron from a full outer valence shell is very difficult due to the amount of energy required to kick the element out of its ideal state (a full outer valence shell).
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