The opposite ring has the same additive group and identity as a ring , with multiplication . Left modules over it are right modules over . Forming the opposite exchanges left and right chain conditions and reverses the order of composition when interpreting endomorphisms as right multiplications. For an associative algebra with its scalar action retained, this construction gives the opposite algebra.
In mathematics, specifically in abstract algebra, an **opposite ring** is a concept that arises when considering the structure of rings in a different way. If \( R \) is a ring, the **opposite ring** \( R^{op} \) (also sometimes denoted as \( R^{op} \) or \( R^{op} \)) is defined with the same underlying set as \( R \), but with the multiplication operation reversed.
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