In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of R). With the initial definition (which is used in this article), the only ideal of R that is a subring of R is R itself.

Podzbiór  pierścienia  nazywamy jego podpierścieniem, jeżeli jest on pierścieniem ze względu na te same działania, co pierścień .

В абстрактной алгебре, области математики, подкольцо — это подмножество кольца, содержащее мультипликативную единицу и само являющееся кольцом относительно тех же бинарных операций.