Preliminaries

By a semiring (S; +, •) we understand a general algebra with two binary associative operations fulfilling the following distributive laws: 1 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWG4bGaai4Tamaabmaapaqaa8qacaWG5bGaey4kaSIaamOEaaGa % ayjkaiaawMcaaiabg2da9iaadIhacaGG3cGaamyEaiabgUcaRiaadI % hacaGG3cGaamOEaaaa!4528! $$x\cdot \left( {y + z} \right) = x\cdot y + x\cdot z$$ and 2 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaqadaWdaeaapeGaamyEaiabgUcaRiaadQhaaiaawIcacaGLPaaa % caGG3cGaamiEaiabg2da9iaadMhacaGG3cGaamOEaiabgUcaRiaadQ % hacaGG3cGaamiEaaaa!452A! $$\left( {y + z} \right)\cdot x = y\cdot z + z\cdot x$$ for all x, y, z ∈ S. If the addition is commutative and has a neutral element 0 (i.e. if (S; +) is a commutative monoid), which is an annihilating (or absorbing) element, that 3 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWG4bGaey4kaSIaamyEaiabg2da9iaadMhacqGHRaWkcaWG4baa % aa!3CD4! $$x + y = y + x$$ 4 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWG4bGaey4kaSIaaGimaiabg2da9iaadIhacqGH9aqpcaaIWaGa % ey4kaSIaamiEaaaa!3E4F! $$x + 0 = x = 0 + x$$ 5 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWG4bGaai4TaiaaicdacqGH9aqpcaaIWaGaeyypa0JaaGimaiaa % cElacaWG4baaaa!3EBE! $$x\cdot 0 = 0 = 0\cdot x$$ for arbitrary x, y ∈ S, then (S; +, •) is said to be a hemiring (see the monographs by J.S. Golan [1992], [and such papers as K. Iizuka [1959], D.R. LaTorre [1965], D.M. Olson [1978], D.M. Olson & T.L. Jenkins [1983] and S.M. Yusuf & M. Shabir [1988]; note that there are some differences in terminology). If a semiring S has an element 0 with property (4), then S is called a semiring with zero. A semiring with zero and commutative addition is called (by some theoretical physicists and categorists) a rig (in the sense of “rings without negation”). Observe that, in general, the zero element does not need to be annihilating (see, e.g., K. Głazek [1968a]).