I have a basic question on homotopy theory, and I would welcome answers or references both from the classic and the motivic context of homotopy theory.

Let $\mathbb{E}=(E_n)_{n\in \mathbb{N}}$ be an $\Omega$-spectrum which is also a ring spectrum. That is to say, there is a map $\mu\colon \mathbb{E} \wedge \mathbb{E}\to \mathbb{E}$ satisfying the usual properties so that the cohomology defined by $\mathbb{E}$ has a cup-product $$ \mathbb{E}^{p}(\underline{\phantom{a}})\otimes \mathbb{E}^q (\underline{\phantom{a}})\to \mathbb{E} ^{p+q}(\underline{\phantom{a}}) \ $$ A fact from classic homotopy theory is that $E_0=\Omega^\infty \mathbb{E}$ is also a ring object (since the functor $\mathbb{E}^{0}(\underline{\phantom{a}})$ maps spaces to rings). Hence, there is a map $u\colon E_0\times E_0\to E_0$ with usual properties representing the cup-product $\mathbb{E}^{0}(\underline{\phantom{a}})\otimes \mathbb{E}^0 (\underline{\phantom{a}})\to \mathbb{E} ^{0}(\underline{\phantom{a}})$. Since $E_n=\Omega^\infty \mathbb{E}[n]$ represents the groups $\mathbb{E}^n(\underline{\phantom{a}})$, my question is the following:

Question:$$\mbox{Are there maps } u_{p,q}\colon E_p\times E_q\to E_{p+q} \,, \, p,q\in \mathbb{N}\, ,\, \mbox{ inducing the cup product}? $$ $$\mbox{Maybe maps from } E_p\wedge E_q \ \mbox{ to } \ E_{p+q}?$$