After discussing how the Hebrew word aluf came to mean general (of the army), let's look at the Hebrew word for admiral (of the navy). This one's easy - אדמירל. The transliteration is the official definition, since the Israeli navy is part of the IDF, and there does not seem to be an Israeli position equivalent to the US admiral (see this site: Rank Insignia of the Israeli Navy with comparison to the US Navy and the Royal Navy for more detail.)
However, our search for Semitic roots does not end here. The English word admiral derives from Arabic. The Online Etymological Dictionary writes:
from Arabic title amir-ar-rahl "chief of the transport," officer in the Mediterranean fleet, from amir "leader," influenced by L. ad-mirabilis (see admire).
The 1911 Encyclopedia gives a slightly different origin:
ADMIRAL, the title of the general officer who commands a fleet, or subdivision of a fleet. The origin of the word is undoubtedly Arabic. In the 12th century the Mediterranean states which had close relations with the Moslem powers on the shores or in the islands of that sea, found the title amir or emir in combination with other words used to describe men in authority; the amir-al-mumenin prince of the faithful or amir-al-bahr commander of the sea. They took the substantive " amir " and the article " al " to form one word, " amiral " or " ammiral" or "almirante." The Spaniards made mirama-molin, out of amiral-mumenin, in the same way. "Amiral," as the name of an eastern ruler, became familiar to the northern nations during the crusades. Layamon, writing in the early years of the 13th century, speaks of the "ammiral of Babilon," and the word was for long employed in this sense. As a naval title it was first taken by the French from the Genoese during the crusade of 1249. By the end of the 13th century it had come to be used in England as the name of the officer who commanded the Cinque Port ships. The English form "admiral " arose from popular confusion with the Latin admirabilis.
I'm not sure if either rahl (transport) or bahr (sea) have Hebrew cognates. (Bahr is however the source of the name of the country Bahrain - "two seas". ) However the words emir or amir certainly have a Hebrew connection: they are related to the Hebrew אמר "he said", via the Arabic amara "he commanded".