"act of rivaling, competition, strife or effort to attain an object another is pursuing," 1590s; from rival (n.) + -ry. Shakespeare has rivality ("Antony and Cleopatra"), but meaning "association, partnership, equality in rank," from the secondary sense of the Latin adjective. Jonson has rivalship (1630s); rivaltry (1640s) also was used.
word-forming element making adjectives from nouns, meaning "having, full of, having to do with, doing, inclined to," from Old French -ous, -eux, from Latin -osus (compare -ose (1)). In chemistry, "having a lower valence than forms expressed in -ic."
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Harper, D. (n.d.). Etymology of rivalrous. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/rivalrous