on the coast
"I have a summer house on the coast"
is 'на море' wrong?

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Answer

Можно еще сказать - на побережье, на берегу моря.

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No, it's right.
"У меня есть летний домик на море" - it means that the house is next to the sea

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Yes, "на море" is wrong expression in this case. Athough if you can say about your holiday on a seashore you may use the phrase "Проводить отпуск на море".

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me too

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Моя голубая мечта - на старости лет, завести домик на Чёрном море.